LIBRARY OF CONGRESS. 



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UNITED STATES OF AMERICA. 



WORKS OF PROF. W. W. JOHNSON 

w 

PUBLISHED BY 

JOHN WILEY & SONS. 



An Elementary Treatise on the Integral Calculus. 

Founded on the Method of Rates or Fluxions. A 
companion book to Rice and Johnson's Abridged 
Differential Calculus. By Prof.' W. W. Johnson, 
U. S. Naval Academy, with the co-operation of 
Prof. J. M. Rice, U, S. Naval Academy. 12030, 
cloth, $1.50. 

The Theory of Errors and the Method of Least 
Squares. 

i2mo, cloth, $1.50. 
Curve Tracing in Cartesian Coordinates. 

i2mo, cloth, $1.00. 
Differential Equations. 

A Treatise on Ordinary and Partial Differential 
Equations. 8vo, cloth, $3-50. 



AN 



ELEMENTARY TREATISE 



INTEGRAL CALCULUS 



FOUNDED ON THE 



METHOD OF RATES OR FLUXIONS 



WILLIAM WOOLSEY JOHNSON 

PROFESSOR OF MATHEMATICS AT THE UNITED STATES NAVAL ACADEMY 
ANNAPOLIS MARYLAND 



REVISED EDITION 
FIRST THOUSAND. 



NEW YORK 

JOHN WILEY & SONS 

London: CHAPMAN & HALL, Limited 

1898 




^ 






i;i9o 



Copyright, 1898, 

BY 

JOHN WILEY & SONS. 




TWO COPIES RECEIVED* 



CRT D*WI 



ROBERT DRWMMOND, PRINTER, NEW YORK. 



■ 



CONTENTS. 

CHAPTER I. 

Elementary Methods of Integration. 

I. 

PAGE 

Integrals - . . I 

The differential of a curvilinear area 3 

Definite and indefinite integrals 4 

Elementary theorems 6 

Fundamental integrals 7 

Examples I io 

II. 

Direct integration * . < . . 14 

Rational fractions 15 

Denominators of the second degree 16 

Denominators of degrees higher than the second 19 

Denominators containing equal roots , 21 

General expression for the numerator of a partial fraction 22 

Examples II 26 

III. 

Trigonometric integrals 33 

Cases in which sin w cos* e de is directly integrable 34 

The integrals sin 2 eafe, and cos 2 0d<9 ,.....*.'. 36 

The integrals [ — *_ , fj*L , and f _*_ 37 

J sin e cos e J sin e J cos e 

Hi 



IV CONTENTS. 

PAGE 

Miscellaneous trigonometric integrals 38 

de 

The integration of : 40 

& a 4- b cos 9 * 

Examples III 43 



CHAPTER II. 

Methods of Integration — Continued. 

IV. 

Integration by change of independent variable 50 

Transformation of trigonometric forms 51 

Limits of a transformed integral 53 

The reciprocal of x employed as the new independent variable 53 

A power of x employed as the new independent variable 55 

Examples IV 56 

V 

Integrals containing radicals 59 

Radicals of the form ^/(ax 2 + b) 61 

The integration of 64 

i/(x 2 ± a'') 

Transformation to trigonometric forms 65 

Radicals of the form ^/(ax 2 4- bx + c) , 67 

The integrals I ^ and f — 68 

Examples V 70 

VI. 

Integration by parts . 77 

Geometrical illustration 78 

Applications 78 

Formulae of reduction 81 



Reduction of I sin w 9 dB and cos w Q dQ 82 

Reduction of |sin w 9 cos* 9 dQ - 84 



CONTENTS. V 

PAGE 

Illustrative examples 87 

Exa?nples VI 89 



VII. 

The integral and its limits 95 

Graphic representation of an integral 99 

Multiple-valued integrals 102 

Formulae of reduction for definite integrals 106 

Change of independent variable in a definite integral 109 

The definite integral regarded as the limiting value of a sum 1 11 

The differentiation of an integral 114 

Integration under the integral sign 116 

Additional formulse of integration 119 

Examples VII 121 

CHAPTER III. 

Geometrical Applications — Double and Triple Integrals. 

VIII. 

Areas generated by variable lines having fixed directions 127 

Application to the witch 128 

Application to the parabola when referred to oblique coordinates 130 

The employment of an auxiliary variable 130 

Areas generated by rotating variable lines 132 

The area of the lemniscata 133 

The area of the cissoid 134 

A transformation of the polar formulge , 134 

Application to the folium 135 

Examples VIII 138 

IX. 

The volumes of solids of revolution 145 

The volume of an ellipsoid , 147 

Solids of revolution regarded as generated by cylindrical surfaces 148 

Examples IX i 149 

X. 

Double integrals 153 

Limits of the double integral 154 

The area of integration 157 



VI CONTENTS. 



PAGE 

Change of the order of integration 15& 

Triple integrals 160 

Examples X 162 

XL 

The polar element of area 164 

Cylindrical coordinates 165 

Solids of revolution with polar coordinates 167 

Polar coordinates in space 168 

Volumes in general 170. 

Examples XI 172 

XII. 

Rectification of plane curves 174 

Rectification of the semi-cubical parabola 174 

Change of sign of ds 176 

Polar coordinates 176 

Rectification of curves of double curvature 177 

Rectification of the loxodromic curve 178 

Examples XII 179 

XIII. 

Surfaces of solids of revolution 184 

Quadrature of surfaces in general 185, 

The expression in partial derivatives for sec y 187 

The determination of surfaces by polar formulae. 188 

Examples XIII 189 

XIV. 

Areas generated by straight lines moving in planes 192 

Applications , . , 193 

Sign of the generated area * „ 195 

Areas generated by lines whose extremities describe closed circuits 196 

Amsler's Planimeter 197 

Examples XIV 199 

XV. 

Approximate expressions for areas and volumes 201 

Simpson's rules 203 



CONTENTS. Vll 



PAGE 

Cotes' method of approximation , , . 204 

Weddle's rule t . 204 

The five-eight rule 205 

The comparative accuracy of Simpson's first and second rules 206 

The application of these rules to solids 206 

Woolley's rule 207 

Examples XV 208 



CHAPTER IV. 

Mechanical Applications. 

XVI. 

Definitions 210 

Statical moment 210 

Centres of gravity .. 212 

Polar formulae 214 

Centre of gravity of the lemniscata 215 

Solids of revolution „ 215 

Centre of gravity of a spherical cap 216 

The properties of Pappus 216 

Examples XVI 218 

XVII. 

Moments of inertia 225 

Moment of inertia of a straight line 226 

Radii of gyration, , 226 

Radius of gyration of a sphere 227 

Radii of gyration about parallel axes 228 

Application to the cone 229 

Polar moments of inertia 231 

Examples XVII 23 1 



VUl CONTENTS. 



PAGE 

i he determination of volumes by triple integration 150 

Elements of area and volume ICl2 

Polar elements j -. 

The determination of volumes by polar formulas 155 

Polar coordinates in space I cn 

Application to the volume generated by the revolution of a cardioid 159 

Examples IX „ jfo 



X. 

Rectification of plane curves .... 168 

Rectification of the semi-cubical parabola 168 

Rectification of the four-cusped hypocycloid 169 

Change of sign of ds 1 70 

Polar coordinates 170 

Rectification of curves of double curvature 171 

Rectification of the loxodromic curve 172 

Examples X 173 

XI. 

Surfaces of solids of revolution 178 

Quadrature of surfaces in general 179 

The expression in partial derivatives for sec 7 i&o 

The determination of surfaces by polar formulas 181 

Examples XI 183 

XII. 

Areas generated by straight lines moving in planes 186 

Applications 187 

Sign of the generated area 189 

Areas generated by lines whose extremities describe closed circuits 190 

Amsler's Planimeter 191 

Examples XII 193 



XIII. 

Approximate expressions for areas and volumes . . . . . I95 

Simpson's rules 197 

Cotes' method of approximation . . , 198 



CONTENTS. IX 



PAGB 

Weddle's rule 199 

The five-eight rule 199 

The comparative accuracy of Simpson's first and second rules '. . . 200 

The application of these rules to solids , 200 

Woolley's rule 201 

Examples XIII ... 202 



CHAPTER IV. 

Mechanical Applications. 
XIV. 

Definitions 204 

Statical moment 204 

Centres of gravity , 206 

Polar formulas 208 

Centre of gravity of the lemniscata 209 

Solids of revolution 209 

Centre of gravity of a spherical cap 210 

The properties of Pappus 210 

Examples XIV 212 

XV. 

Moments of inertia 219 

Moment of inertia of a straight line 220 

Radii of gyration 220 

Radius of gyration of a sphere 221 

Radii of gyration about parallel axes 222 

Application to the cone 223 

Polar moments of inertia 225 

Examples XV 225 



THE 

INTEGRAL CALCULUS 



CHAPTER I. 

Elementary Methods of Integration. 



I. 

Integrals, 

I. In an important class of problems, the required quanti- 
ties are magnitudes generated in given intervals of time with 
rates given in terms of the time /; or else, being assumed to 
be so generated concurrently with some other independent 
variable, have rates expressible in terms of this independent 
variable and its rate. 

For example, the velocity of a freely falling body is known 
to be expressed by the equation 

*>=£?> (i) 

in which / is the number of seconds which have elapsed since 
the instant of rest, and g is a constant which has been deter- 
mined experimentally. If s denotes the distance of the body 



ELEMENTARY METHODS OF INTEGRATION. [Art. I, 



at the time t, from a fixed origin taken on the line of motion, 
v is the rate of s ; that is, 

ds 
V = * ; 

hence equation (i) is equivalent to 

ds — gtdt, (2) 

which expresses the differential of s in terms of t and dt. Now 
it is obvious that \gp is a function of t having a differential 
equal to the value of ds in equation (2) ; and, moreover, since 
two functions which have the same differential (and hence the 
same rate) can differ only by a constant, the most general 
expression for s is 

s = igf+C, (3) 

in which C denotes an undetermined constant. 

2. A variable thus determined from its rate or differential 
is called an integral, and is denoted by prefixing to the given 

differential expression the symbol , which is called the integral 

sign.* Thus, from equation (2) we have 



\gtdt, 



which therefore expresses that s is a variable whose differential 
is gtdt ; and we have shown that 



\gtdt = \gfi+ C. 



The constant C is called the constant of integration ; its 
occurrence in equation (3) is explained by the fact that we 
have not determined the origin from which s is to be measured. 



* The origin of this symbol, which is a modification of the long s, will be 
explained hereafter. See Art. 9^. 



§ L] THE DIFFERENTIAL OF A CURVILINEAR AREA. 3 

If we take this origin at the point occupied by the body when 
at rest, we shall have s = o when / = o, and therefore from 
equation (3) C=0', whence the equation becomes s = \gft. 



The Differential of a Curvilinear Area, 

3. The area included between a curve, whose equation is 
given, the axis of x and two ordinates affords an instance of 
the second case mentioned in the first paragraph of Art. 1 ; 
namely, that in which the rate of the generated quantity, al- 
though not given in terms of t, can be readily expressed by means 
of the assumed rate of some other 
independent variable. 

Let BPD in Fig. 1 be the curve 
whose equation is supposed to be 
given in the form 

Supposing the variable ordinate 

PR to move from the position AB 

to the position CD, the required 

area A B DC is the final value of the Fig. i. 

variable area ABPR, denoted by 

A, which is generated by the motion of the ordinate. The rate 

at which the area A is generated can be expressed in terms of 

the rate of the independent variable x. The required and the 




assumed rates are denoted, respectively, by 



dA , dx , L 

— - — and — - - ; and, to 
dt dt 



express the former in terms of the latter, it is necessary to 
express dA in terms of dx. Since x is an independent variable, 
we may assume dx to be constant ; the rate at which A is gen- 
erated is then a variable rate, because PR or y is of variable 
length, while R moves at a constant rate along the axis of x. 
Now dA is the increment which A would receive in the time 



4 ELEMENTARY METHODS OF INTEGRATION. [Art. 3. 

dt t were the rate of A to become constant (see Diff. Calc, 
Art. 17). If, now, at the instant when the ordinate passes the 
position PR in the figure, its length should become constant, 
the rate of the area would become constant, and the increment 
which would then be received in the time dt, namely, the 
rectangle PQSR, represents dA. Since the base RS of this 
rectangle is dx, we have 

dA — ydx — f (x)dx (1) 

Hence, by the definition given in Art. 2, A is an integral, and 
is denoted by 



A = 



\fW* (*) 



Definite Integrals. 

4. Equation (2) expresses that A is a function of x, whose 
differential is f(x)dx ; this function, like that considered in Art. 
2, involves an undetermined constant. In fact, the expres- 
sion f(x)dx is manifestly insufficient to represent precisely 

the area ABPR, because OA, the initial value of x, is not indi- 
cated. The indefinite character of this expression is removed 
by writing this value as a subscript to the integral sign ; thus, 
denoting the initial value by a, we write 



A 



= ^f(x)dx, (3) 



in which the subscript is that value of x for which the integral 
has the value zero. 

If we denote the final value of x (OC in the figure) by &, the 
area ABDC, which is a particular value of A, is denoted by 



§ I.] DEFINITE INTEGRALS. 5 

writing this value of x at the top of the integral sign, 
thus, 

'b 



ABDC = 



f(x)dx (4) 



This last expression is called a definite integral, and a and 
b are called its limits. In contradistinction, the expression 

f(x)dx is called an indefinite integral. 

5. As an application of the general expressions given in the 
last two articles, let the given curve be the parabola 

y — x 2 . 
Equation (2) becomes in this case 

A - [ x*dx. 

Now, since \x* is a function whose differential is x 2 dx, this 
equation gives 

A = [ x 2 dx = \x* + C, (1) 

in which C is undetermined. 

Now let us suppose the limiting ordinates of the required 
area to be those corresponding to x — 1 and x — 3. The vari- 
able area of which we require a special value is now represented 

by I x 2 dx, which denotes that value of the indefinite integral 

which vanishes when x = 1. If we put x = 1 in the general 
expression in equation (1), namely \x* + C, we have \ + C; 
hence if we subtract this quantity from the general expression, 
we shall have an expression which becomes zero when x = I. 
We thus obtain 

A = [ x*dx = ix* - |. 



ELEMENTARY METHODS OF INTEGRATION. [Art. 5 V 



Finally, putting, in this expression for the variable area, x = 3, 
we have for the required area 

6. It is evident that the definite integral obtained by this 
process is simply the difference between the values of the indefinite 
integral at the upper and lower limits. This difference may be 
expressed by attaching the limits to the symbol ] affixed to the 
value of the indefinite integral. Thus the process given in the 
preceding article is written thus, 



x^dx = -^x 3 + C\ = 9 — 



The essential part of this process is the determination of 
the indefinite integral or function whose differential is equal to 
the given expression. This is called the integration of the 
given differential expression. 

Elementary Theorems. 

7. A constant factor may be transferred from one side of the 
integral sign to the other. In other words, if m is a constant 
and u a function of x, 



mudx = m udx. 



Since each member of this equation involves an arbitrary 
constant, the equation only implies that the two members have 
the same differential. The differential of an integral is by 
definition the quantity under the integral sign. Now the 
second member is the product of a constant by a variable 

factor ; hence its differential ismdl \udxV that is, m u dx t which 

is also the differential of the first member. 



§ I.] ELEMENTAL Y THEOREMS. 7 

8. This theorem is useful not only in removing constant 
factors from under the integral sign, but also in introducing 
such factors when desired. Thus, given the integral 

\x n dx; 

recollecting that 

d(x M + I ) = (n + i)x n dx, 

we introduce the constant factor n + i under the integral sign ; 
thus, 



x n dx 



n + i 



(n + i)x n dx = — — x n + T + C. 
v ' n + I 



9. If a differential expression be separated into parts, its in- 
tegral is the sum of the integrals of the several parts. That is, 
if u> v, w, - - • are functions of x, 

\{u + v + w + • • • -)dx — \udx-\-\vdx+\wdx+ • 

For, since the differential of a sum is the sum of the differ- 
entials of the several parts, the differential of the second mem- 
ber is identical with that of the first member, and each member 
involves an arbitrary constant 

Thus, for example, 

(2 — Vx) dx = 2dx — \x dx = 2x — \x + C, 

the last term being integrated by means of the formula deduced 
in Art. 8. 

Fundamental Integrals. 

10. The integrals whose values are given below are called 
the fundamental integrals. The constants of integration are 
generally omitted for convenience. 



8 ELEMENTARY METHODS OF INTEGRATION. [Art. IO. 

Formula (a) is given in two forms, the first of which is de- 
rived in Art. 8, while the second is simply the result of putting 
n~ — m. It is to be noticed that this formula gives an indeter- 
minate result when n — — I ; but in this case, formula (b) may 
be employed.* 

The remaining formulas are derived directly from the for- 
mulas for differentiation; except that (/'), (£'), (/'), and (m') 

x 
are derived from (j), (k), (/), and (m) by substituting - for x. 

[ , x n + 1 [dx I 

\x n dx = — =—-7 v . . . (a) 

J n + I )x m (m — i) x m ~ x 

J^=log(±*)t • • iP) 

\ aXdX = \^a h* = B * W 

Jcos0^0 = sin0 (d) 

\smBd6 = - cos 6 (e) 

[b 

* Applying formula {a) to the definite integral x n dx, we have 

)a 

)b h n + l „ n + 1 
x dx = _ , 
a »+ I 

which takes the form - when n = — I ; but, evaluating in the usual manner, 

= 2 5_ = log b — log a ; 

a result identical with that obtained by employing formula (b). 

f That sign is to be employed which makes the logarithm real. See Diff, Calc. ; 
Art. 43. 



§ I.] FUNDAMENTAL INTEGRALS. 9 

JcoW = j secW * = tan e * ■ • : ; • • v; • - ! CO 

(^ = f cosec ^^ = - cot ^ ^ 

f sin ^f * = f secfl tan <# = sec 6 ....... (h) 

J cos^ J v J 

— r-o-Tr- = cosec 6 cotd dd = — cosec 0. . . . . (Y) 
J siir ) 

\~V(T^1?) = sin_1 * + c = - cos_1 * + C ' • • ■ C/) 

[ f* =sin- I -+ C^-cos- 1 -+ l' . . . (>') 

J V(a 2 — X s ) a a w J 

f dx _ = tare 1 * + C=- cot" 1 * + C (k) 

Ji + x 2 

f^_^ = Itan- 1 -+ C= - -cot- 1 -+ C. . . . (£') 
Jar + ;r # a a a 

— -rr^ -= sec 1 * + C— — cosec" 1 * + C . . (/) 

JxV(x 2 — i) v ' 

— TT-o 2V = - sec x -+ dT = cosec * - 4- C . (I) 

J * V(x* — a 2 ) a a a a v J 

-77 =r- = vers -1 * . . (m) 

J 4/(2* — X s ) v ' 

f dx _ t x , A 

—77 or- = vers 1 - (#z) 

J V(2tf* — *2) tf V ' 



IO ELEMENTARY METHODS OF INTEGRATION, [Ex. I, 

Examples I. 

Find the values of the following integrals : 
[dx 

[dx i 

J x X 

I dx 2 

3 -J.^' 2 —vx- 

4 - J, Z' *• 

6. f (x-iydx, ?l-. x * + x -l 
J 1 3 

a a 

[~b . . V2 , 1 ' r . ^VlT « 3 

7. {a — bx) dx, ax — abx + = — . 

Jo 3 Jo zo 

(a 22 4 — 1 (i 

{a + x)*dx, a z x + ^^- + ax 3 +- \ = 4a 4 . 

-a 2 4_l-« 

9. j^ — , 2log«. 

10. j — , l0g(_^)J =log2. 



I.] 



EXAMPLES. 



II 



( 4a (a + a 



x)> 



dx, 



f 

12. f^^/x, 

13. sin Odd, 

•* o 

14. COS .*</.#, 

J o 



15 



16 



•i: 

' J V(^_^)> 



4/*(a 2 + fa* + i* 2 )l = 23H • f . 



1 — COS #. 

• 1" 

mx = o. 

"llir 



tan# 



17 



18. 



f 00 <fo 
J- eo a +.* a, 



d&e 



, * V (* 2 - 1) 



sin- 1 - =- 
tfjo 6 



19. If a body is projected vertically upward, its velocity after / units 
of time is expressed by 

v = a — gt, 

a denoting the initial velocity ; find the space s x described in the time 
/i and the greatest height to which the body will rise. 



h = \ v dt = at x — \gt?, 



when v= o , / = — , i" = — 



12 ELEMENTARY METHODS OF INTEGRATION. [Ex. I. 

20. If the velocity of a pendulum is expressed by 

rtt 

v = a cos — , 

the position corresponding to t = o being taken as origin, find an ex- 
pression for its position s at the time /, and the extreme positive and 
negative values of s. 

2Ta . 7lt 

s = sm — , 

7t 2T 

s = ± — - when t=r, 37-, 51-, etc. 

21. Find the area included between the axis of x and a branch of 
the curve 

y = sin x. 2. 

22. Show that the area between the axis of x, the parabola 

f = 4<zx, 

and any ordinate is two thirds of the rectangle whose sides are the 
ordinate and the corresponding abscissa. 

23. Find (<*) the area included by the axes, the curve 

and the ordinate corresponding to x = 1, and (/3) the whole area be- 
tween the curve and axes on the left of the axis of y. 

(a) 8 - i, (ft) 1. 

24. Find the area between the parabola of the #th degree, 

a n ~ I y = x n 9 

and the coordinates of the point (a, a). 



n + 1 



§ I.] EXAMPLES. 13 

25. Show that the area between the axis of x, the rectangular 

hyperbola 

xy= 1, 

the ordinate corresponding to x = 1, and any other ordinate is 
equivalent to the Napierian logarithm of the abscissa of the latter 
ordinate. 

For this reason Napierian logarithms are often called hyperbolic 
logarithms. 

26. Find the whole area between the axes, the curve 

fx™ — a n + m , 
and the ordinate for x — a, m and n being positive. 

If n > m. 



na 

n — m 



if n 5 m, 00 . 

27. If the ordinate BR of any point B on the circle 

x 2 +/=a* 

be produced so that BR • RP — a 2 , prove that the whole area between 
the locus of P and its asymptotes is double the area of the circle. 

28. Find the whole area between the axis of x and the curve 

y ( a 2 + x*) = a\ 

na*. 

29. Find the area between the axis of x and one branch of the com- 
panion to the cycloid, the equations of which are 

x = a*p y = a (1 — costp). 



14 ELEMENTARY METHODS OF INTEGRATION. [Art. II. 

II. 

Direct Integration. 

II. In any one of the formulas of Art. 10, we may of course 
substitute for x and dx any function of x and its differential. 
For instance, if in formula (b) we put x — a in place of x, we 
have 

f dx 

J ^~^a = l ° g (* ~ *) ° r log ^ ~ *)' 

according as x is greater or less than a. 

When a given integral is obviously the result of such a sub- 
stitution in one of the fundamental integrals, or can be made 
to take this form by the introduction of a constant factor, it is 

said to be directly integrable. Thus, sin m x dx is directly in- 

tegrable by formula (e) ; for, if in this formula we put mx for 0, 
we have 



i 



sin mx • mdx = — cos mx , 
hence 



sin mx dx — — sin m x ■ m dx = — — 
J m) n> 

So also in V{a + bx*) x dx , 



cos mx, 
m 



the quantity x dx becomes the differential of the binomial 
(a + dx 2 ) when we introduce the constant factor 26, hence this 
integral can be converted into the result obtained by putting 



(a + bx 2 ) in place of ^in l^/xdx, which is a case of formula (a). 
us 
V (a + bx*)xdx = JL j ( a + bx*f 2bx dx = ~(a + bx*f . 



Thus 



•§11.] DIRECT INTEGRATION. 1 5 

12. A simple algebraic or trigonometric transformation 
sometimes suffices to render an expression directly integrable, 
or to separate it into directly integrable parts. Thus, since 
— sin x dx is the differential of cos x, we have by formula (b) 

sin x dx , 

— log cos x . 



tan x dx 



cos x 
So also, by formula (/), 

[tan 2 6d6= (sec 2 6- i) dB = tan 6 - 6 ; 
by (e) and (a), 

[sin 3 6 dd = f (i - cos 2 6) sin Odd = - cos 6 + i cos 3 6 
by (» and (a), 

dx 



^—^ - M(i - «■)"* (- 2* afc) = sin" 1 * - V (i - **). 



Rational Fractions. 

13. When the coefficient of dx in an integral is a fraction 
whose terms are rational functions of x, the integral may gen- 
erally be separated into parts directly integrable. If the de- 
nominator is of the first degree, we proceed as in the following 

example. 

Given the integral —dx: 

by division, 



2* + I 



&-x + 3 = x 3 15 i 

2X + l 2 4 42J+l' 



1 6 ELEMENTARY METHODS OF INTEGRATION. [Art. 13, 



hence 




















\" 


- x + 

2-2" + I 


3 <£tr 


= — \ x dx - 
2) 


-If- 


+ I5 [- 

4 )■ 


dx 

IX + 1 










_x 2 
4 " 


_.3f 
4 


+ ^.o g 


(2X + 


I). 



When the denominator is of higher degree, it is evident that 
we may, by division, make the integration depend upon that of 
a fraction in which the degree of the numerator is lower than 
that of the denominator by at least a unit. We shall consider 
therefore fractions of this form only. 



Denominators of the Second Degree. 

14-. If the denominator is of the second degree, it will (after 
removing a constant, if necessary) either be the square of an 
expression of the first degree, or else such a square increased 
or diminished by a constant. As an example of the first case, 
let us take 

' — -*dx. 

J Kx - if 

The fraction may be decomposed thus : 

x + 1 x — \ + 2 1 2 



(x-if- (x-if - x-i^ (x-if 
hence 

f x + 1 , f dx f dx 



= log (x - I) 



15. The integral f 2 * +3 , . dx 

J X 1 + 2X + 6 



§11.] RATIONAL FRACTIONS. \J 

affords an example of the second case, for the denominator 
may be written in the form 

jtr 2 + 2x + 6 = (x + i) 2 4- 5. 

Decomposing the fraction as in the preceding article, 

x + 3 x + 1 2 

(x + i) 2 + 5 " (-*" + i) 2 + 5 + O + i) 2 + 5 ; 
whence 



f * + 3 dx = \ (*+^* +2 [ d * 

jx 2 + 20; + 6 J(,r + i) 2 + 5 J(* + i) 2 + 



5 



The first of the integrals in the second member is directly 
integrable by formula (b\ since the differential of the denom- 
inator is 2 (x + i)dx, and the second is a case of formula (k r ). 
Therefore 

[ . X + 3 , ak = i log Cr 2 + 2* + 6) + ~ tan" 1 ^^ • 
16. To illustrate the third case, let us take 

2J+ I 

-z.d x > 



_ x 2 — X 

in which the denominator is equivalent to (x — \) 2 — 6J, and 
can therefore be resolved into real factors of the first degree. 
We can then decompose the fraction into fractions having these 
factors for denominators. Thus, in the present example, as- 
sume 

2* + 1 A B , - 

+ ^7T^' .... (1) 



x — 6 x — 3 .r + 2 



in which A and i? are numerical quantities to be determined. 
Multiplying by (x — 3) (x + 2), 

2jt + I = A (x + 2) + £ (jit — 3). . . . . (2) 



1 8 ELEMENTARY METHODS OF INTEGRATION. [Art. l6. 



Since equation (2) is an algebraic identity, we may in it assign 
any value we choose to x. Putting x = 3, we find 

7 = $A, whence A = -J, 

putting x — — 2, 

— 3 = — $B> whence B = ■§. 

Substituting these values in (1), 

2* + 1 == 7 3 

^-^-6 50-3) 5(* + 2)' 
whence 

2X + I 
J_ x _ 6 d * = % l °8 (* ~ 3) + I ^g (* + 2). 

17. If the denominator, in a case of the kind last considered, 
is denoted by (x — a) (x — b), a and b are evidently the roots of 
the equation formed by putting this denominator equal to zero. 
The cases considered in Art. 14 and Art. 15 are respectively 
those in which the roots of this equation are equal, and those 
in which the roots are imaginary. When the roots are real and 
unequal, if the numerator does not contain x, the integral can 
be reduced to the form 

f dx 

)(x-a)(x-bY 

and by the method given in the preceding article we find 

\ {x _ a)\ X - b) = T=Tb [ l0g {X - a) ~ bg ( * - *>_ 

log ?Pl, (A)* 



a — b s x — b 1 



* The formulae of this series are collected together at the end of Chapter II. 
for convenience of reference. See Art. 101. 



§11.] DENOMINATORS OF THE SECOND DEGREE. 1 9 

in which, when x < a, log (a — x) should be written in place of 
log (x — a). [See note on formula (b), Art. 10.] 
If b — — a, this formula becomes 

f dx \ x — a A 

Integrals of the special forms given in (A) and (A') may be 
evaluated by the direct application of these formulas. Thus, 
given the integral 

r dx 

J 2X 1 + ix — 2 ' 

if we place the denominator equal to zero, we have the roots 
a — \, b = — 2; whence by formula (A), 



dx 



dx I i , x — 4 



J 2X 2 + IX — 2 2 )(x — -\) {x + 2) 2 2-J & ^ + 2 ' 

or, since log (2x — i) differs from log (x — -|) only by a con- 
stant, we may write 

f dx \ . 2x — \ 

log 



2^ + 3^ — 2 5 * X + 2 



Denominators of Higher Degree. 

18. When the denominator is of a degree higher than the 
second, we may in like manner suppose it resolved into factors 
corresponding to the roots of the equation formed by placing it 
equal to zero. The fraction (of which we suppose the numerator 
to be lower in degree than the denominator) may now be decom- 
posed into partial fractions. If the roots are all real and un- 
equal, we assume these partial fractions as in Art. 16 ; there 
being one assumed fraction for each factor. 

If, however, a pair of imaginary roots occurs, the factor cor- 



20 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 8. 

responding to the pair is of the form (x — a) 2 + /2 2 , and the 
partial fraction must be assumed in the form 



2' 



Ax + B 
(x - of + ft 

for we are only entitled to assume that the numerator of each 
partial fraction is lower in degree than its denominator (other- 
wise the given fraction which is the sum of the partial fractions 
would not have this property). For example, given 
f x + 3 



dx. 



Assume 



whence 



(**+ 0(*-0 



x + 3 Ax - B C 

■y — t' ' " * V / 



(x 2 + i)(x — i) " X 2 -f I 



X 



x + 2 = (x-i)(Ax + £) + (x 2 + i)C. 
Since in an identical equation the coefficients of the several 
powers of x must separately vanish, the coefficients of x 2 , x 
and x° give, for the determination of A, B and C, the three 
equations 

A + C = o, B-A = i, C-B=$. 

From these we obtain A = — 2, B = — i, C = 2; hence, 
substituting in equation (i), 

X + 3 2 2I+I 

(jit 2 + 1)0 — I) " x — I ~~ JT* + I ' 
therefore 



f -^ + 3 , _ f ^ f 2 xdx [ 

l(x 2 + i)(jt - if* ~ 2 J^r^l ~~ J x* + l ~ J 



dx 



x*+ I 



= 2 log (x- i) - \og(x 2 + i) - tan" 1 *. 

19. The method of determining the assumed coefficients 
illustrated above makes it evident that, the denominator bein« 



§ II.] MULTIPLE ROOTS. 21 

of the ;/th degree, we must assume n of these coefficients • 
because we have to satisfy n equations, derived from the 
powers of x from x n ~ l to x°. 

It is evident that we may take for the denominators of the 
partial fractions any two or more factors of the given denomi- 
nator which have no common factor between themselves, pro- 
vided we assume for each numerator a polynomial of degree 
just inferior to that of the denominator. But, since our present 
object is to separate the given fraction into directly integrable 
parts, when a squared factor such as (x — a) 2 occurs, instead 
of assuming the corresponding partial fraction in the form 

- 2 (which would require to be further decomposed as in 

Art. 14), we at once assume a pair of fractions of the form 



A B 

— a (x — a) 



2 • 



20. We proceed in like manner when a higher power of a 
linear factor occurs. For example, given 



1 



X + 2 

zdx; 



(x- !)%*• + 1) 



we assume 



x + 2 A B CD 

+ T. 1x5 + 7 + 



(x - if(x +i)~(x-iy~ r (x-i)~x-i^x + i' 
whence 

x + 2 = [A + B (x- 1) + C(x - if](x 4- 1) + D{x - i) 3 . (1) 
Putting x— 1, we have Z — ^A .'. A=%. 

Putting x = — I, we have 1 = — SB ,\ D = — £. 

The most convenient way to determine the other coefficients 
is to equate to zero the coefficient of x 3 , and to put x — O. We 
thus obtain 



22 ELEMENTARY METHODS OF INTEGRA T10N. [Art. 20. 

o=C + D, and 2 = A — B + 6" - A 
from which C = ■£, and i? = — J. Therefore 
r # -f- 2 r _ 3 f ^ \ i dx i c dx i C dx 

J (*-i)v+i) " 2 J (x~=t? ~ 4 J(^ry 2 H ~ 8 J x~=i ~ %)7+i 

* i i ^ — i 



21. The fraction corresponding to a simple factor of the 
given denominator may be found, independently of the other 
partial fractions, by means of the expression derived below. 
Denote the given denominator by (p(x), and let it contain the 
simple factor x — a, so that 

<P(x) = (x - a)^(x), (i) 

in which tp{x) does not vanish when x = a. 

Let f[x) denote the numerator (it is not necessary here to 
suppose this to be lower than <f>\x) in degree), and let Q denote 
the entire part of the quotient. Then we may assume 

A*) _ A*) n , A- P 

00) ' {x — a)\:{x) ^ "*" x - a "*" $(*)' 

in which Q and P are in general polynomials. Clearing of 
fractions, 

f(x) = Q(x - a)ip(x) + Aip(x) + P(x - a). 

Putting x = a, we have (since neither P nor Q can become in- 
finite) 

f{a) = A<P(a), whence A =^ y . . . . (2) 

an expression for the numerator A. 

Again, differentiating equation (i), we have 

<f>'(x) = i/>(x) +(x-a)tp\x); 



23 



A = ^ (3) 



§ JI -J PARTIAL FRACTIONS. 

whence, putting x = a, (p\a) == ip(a), the substitution of which 
in equation (2) gives another expression for A, namely, 

Art 

</>\a) 
As an example, let us find the value of 

r * + 2 . 

J X* + 2^ 3 — X* — 2X 

The denominator is the product ^ 2 — i)(jt + 2), and the first 
term of the quotient is obviously x ; hence we assume 

x 5 + 2 ^4 £ (7 Z> 

= * + <H + H + 



2;r ;r x + l x — I x -\- 2 

The coefficient of .r 4 in the equation cleared of fractions gives 
a = — 2. Now forming the fraction 

fix) _ x* + 2 

<p'{x) ~~ 4x s -f 6x 2 — 2^ — 2' 

we may determine A, B, C and Z> in accordance with expres- 
sion (3) by putting therein for x successively o, — I, 1 and — 2. 
Thus A — — 1, B ^= J, C = £, D = 5 ; whence 

jr 5 H- 2 11 1 C 

+ : v + 



X 4 H- 2-T 5 — X 1 — 2X X 2\X -f I) 2{x—\) X + 2 

and 

f ,r 5 + 2 * 2 (* + 2) 5 4 /(^ - 1) 
— i 5 9 ^ = 2* -{- log — — -. 

J X A + 2X 6 — X Z — 2X 2 to ^T 

22. We have seen that the decomposition of a given frac- 
tion into partial fractions presupposes a knowledge of the roots 
of the equation resulting from equating the given denominator 
to zero. In the case of the denominator x n — 1, we can employ 
the expressions for the imaginary roots involving the circular 
functions of certain angles. (Diff. Calc, Art. 191.) In some 
simple cases the factors are expressible in ordinary surds. For 
example, we have 



24 ELEMENTAR Y METHODS OF INTEGRA TION. [Art. 22. 

S 8 ~ I = (* - l)(* + l)(^ 2 + I)(^ 2 -^| 7 2 + I)(^ 2 + * |/2 + I). 

It is to be noticed that when the fraction is a rational func- 
tion of some higher power of x, we may simplify the process 
of decomposition. Thus it is legitimate to assume 

x* A B 

X s — i ~~ x* — I + x A -f- i' 

because, if # = ^ 4 , the numerator and factors of the denominator 
are rational functions of z. The first term could be treated in 
like manner with respect to x 2 \ but not the second, since the 
factors of x* + I involve x as well as x 2 , 

23. We have seen in the preceding articles that the partial 
fractions corresponding to the reai roots, whether single or 
multiple, are directly integrable, and also those corresponding 
to unrepeated imaginary roots. In the following example a 
case of multiple imaginary roots occurs: given 

f a + x 



(a - x){a 2 + x 2 f 



dx. 



It is readily shown that the partial fraction corresponding 

to a — x is — o, N , and the remainder is conveniently 

2a s (a — x) J 

found by subtracting this from the given fraction ; thus 

a 4- x I c& 4- 2a z x — 2a 2 x 2 — x* 



,2\2 



(a — x){d l -f- ;r 2 ) 2 2« 3 (# — x) 2a\a — x)(a 2 -f x 2 ) 

The numerator of this remainder is found to be divisible by 
a — x, thus verifying the process ; and the given integral re- 
duces to 

I , . i f# 3 4- ia 2 x -f ax 2 + ^r 3 
- -& log («-*)+ ^ J ' ^-^ <&• 

The integration of the last term may be effected by a trans- 
formation given later. See Art. 41. 

24. Instead of assuming the partial fractions with undeter- 



§11.] 



RATIONAL FRACTIONS. 



25 



mined numerators, it is sometimes possible to proceed more 
expeditiously as in the following examples : 
Given 



J* 8 (1 + 



x 2 ) 



dx; 



putting the numerator in the form 1 4- x* — x 2 , we have 



*" 



f I dx = f 1 + x* dx _ f . 



dx 



IS-WW*- 



Treating the last integral in like manner, 

[dx 



J**(i- 



(1 +X 2 ) 

I 



dx 



dx 
1? 



ix [ xc 



dx 
x> 



-^-log* + |log(i + ^)=-J^+log ^ l ^\ 

Again, given 

J**(i + xf dx; 
putting the numerator in the form (1 4- xf — 2x — x*, we have 



J* 2 (1 + *) 2 ^ J* 2 ~. 
_ tdx 



2 + x 



x{i +xf 
dx 



dx 



+ 



x(l + x) J(l + #) 



<^r 



\2* 



Hence by equation (A), Art. 17, 



[ dx _ _\ , x I 

J^ 2 (i+^-) 2_ x ^l+X~l-\- 



26 ELEMENTARY METHODS OF INTEGRATION. [Ex. II 

Examples II. 

f dx 

Jdx 
(a-xf 

f xdx 
3 * )a* +x" 

4. V (a 2 — x*) xdx, 
f x * dx 

' ioVia'-xy 
7- f(« 2 + zx^xdx, 
8. (0 + w^) 8 dx, 

9 ' J sin 2 2x' 
10. cos 3 a: sin x dx, 
f cos 6 d$ 

12. sec 3 3 ^ tan 3* d&, 



— log (a — x). 


1 


a — x* 


- log (a' + ■*"). 


a 3 - (a' - x*f 


3 


1 1 a 3 


3 lo g^_^ 3 ^ 


a — V(a 2 — ^ 2 ). 


• (** + 3*> 


24 


(0 + mx)*— a* 


$m 


COt 2X 

• 

2 


I — cos 4 x 


4 


1 2 
— - - cosec 6 . 
2 


sec 3 $x — 1 


9 



i7- cos 



20 



21 



iLj EXAMPLES. 27 

(a mx 
a mx dx. — ; . 

m log# 

14. f (c* - 1) 3 dx, \&* — f £ 2 * + & x — x. 

f / • a ' \» « -r C 1 + 3sin 2 jc) 3 
*5- v 1 + 3 sm x ) sin#cos#dfr, ^ -. 



J o 

18. sec 4 6d9, 

19. tan 3 xdx, — tan 3 * + log cos x. 



tan H — tan 3 9. 
3 



7T IT 

sec 4 # tan # dx. - sec 4 .r | = — . 

Jo 4 Jo 4 

. L/- — -dx> asm -1 - + Wa* — * 2 ). 

J V + x a 



1 — log 2 

2 



2. Mcot 3 e^G, 

4 

3. Li/ <£r, tf(2ax — x*) + avers -1 -, 

J r JC # 

J. , x , cos (a:— 26) 

sm (a — 26) /#, - , 



28 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. 



2 5 



■j. 



cos x dx 
a — b sin x 



26. 



'4 dx 



tan^ 



— log(# — ^sin^) a 



i log 2. 



27. 



(2 dx 
n tan .r ' 



i log 2. 



28 . J; 

4 
29. J 



dfc 



jr log x y 
dx 



30- 






r x' dx 
J ^ B + 1 



3I : ]v(a<-**y 



32. 



33- 



34- 



f dx 



dx 



2 + s x 



\\- 



dx 



35 



|/(2^ 2 - l) ' 

dx 



r dx 

x 2 + x + 1 



X 

log(- log*) J* = -log 



tan" 1 e x . 





1 

3 


tan " V 




1 
2 


sin 1 -j- 


I 


sin' 


.1 *V3 


I 
4/10 


tan" 


.1 ^V5 

4/2 

7t 

4 



_ 1 2X + I ") 1 _ 7T 



. tan 

V3 4/3 



] 

_lo 



3^3 



§11.] EXAMPLES. 29 

36. f-TT — rv cos_1 t- 

AT 

4/(^ 2 — a 2 )— # sec" 1 -. 

3 s - ir^^ ^ Gog 2 -a. 

39- f 4 * ~* + * 3 ^1 4* - i log (* 2 + 1) - tan" 1 *. 

f.* 2 + x 4- I , 1/2 \ 2 1 2JC — 1 

4'- Jf^i* 5 



r .r — 2 

.* +-log — — 

4 b x + 2 



f(i + •*)' , * 

f(2A* + I) 2 dx a , 1 / , \ 

43- I — ;^r— : — 1 * 2 - * + 2 log (2X + 3). 

(»r + ,)■ *> -log(«+i) 

6 r____*L__ 

J o ^ — 20.x- cos a + # a ' 

1 . x — a cos an* n — a 

*—. tan l ; = ; — 

a sm a a sin a 2a sin ot 



2X + I 

X — 2 
X — I 



30 ELEMENTARY METHODS OF INTEGRATION, [Ex. II, 



47- 



48. 



49. 



5°- 



5i 



5 2 - 



53- 



y 



dx 



= 



2^^ sec <z + 



log 



Jf — a sec <* — a tan <* 



4^-7 



1, 



v 2 ^r 



3-* 






dfc, 



(* + 2)(x + 3 y 

x dx 

x ir + x' 2 + x + I ' 



* 4 + * 2 



f # 2 — ^ -f 2 . 
54- \ X *_ S J +A *<> 



55- 



56. 



57- 



dx 



$x + 1 



dx, 



J 1 + x 



20 tan <z ° x — a sec or + tan « 



4/2 . 2.T — 2 — 3 4/2 

log ■ , . , 

12 ° 2^ — 2 + 34/2 



I . I + ^ 3 

6 log F^7< 



I , x 3 (x — 2) 
6 l0g (* + !)' 



2 log 



^_L3 ___3_ 
^ + 2 x + 3 



L tan " + log ^TT J J 



i, x — 1 4/2^ _, # 
-log — — + - !L - tan 
6 *x + 1 3 



V2 ' 



2, -JC+L, I. X — 2 
- log ; 4" ~ log 

3 & x+2 3 & *-i 



I . # + I 

-log 
4 



JC — I 2(# — i) 



(x — 1) V(x + 1) _ I 
(x 2 + 1)* 2 



log 



log 

(*+ 1) 



I , 2X - I 

+ — r— tan 



' 6 & ^ 2 — ^ + 1 4/3 ^3 



§11.] 



EXAMPLES. 



59- 



dx 



x{i + x + x 2 + X*) 



log x log (1 + x) log ( 1 4- x 2 ) tan - * x. 



60. 



x — I 



x* + x 2 + I 



dx, 



1 ^ — x + 1 

2 l0g *' + X + I 



6l - \ ** + £-l x dx > ^log* + ^log(*- 2) + ^log(* + 3). 
f x 2 dx 

62. — j , 

J X — X — 1 2 

6 3- j(^r7y» 

6 4- J x »_^ <*». 

6 *-h 



\ . x — 2 4/3 x 

-log — ■ — + -^-tan- 1 -— . 
7 & x + 2 7 1/3 

1 , # — 1 x 

— lOSf : . 

4 x + 1 2 (je a — 1 ) 

5 ^ 1 . x — a 

—tan - x — log — ■ — . 

2a a 4a x + a 



x dx 



— x 






fdx 1 x ~\~ b b t 

67 ' Jo(*" + a 2 ){x 2 +b 2 Y 

A* r a * dx 

)ax 2 (a 2 +x 2 )> 
6 9- \—7 — T—^dx* 



2ab (a + b)' 
4 — 7t 



tan- 1 .* + log 



V(i + <T 



lo. 



dx 



I I 

tan -1 * ^ -|. 

x 3^ 



32 ELEMENTARY METHODS OF INTEGRATION. [Ex. II, 



f dx 

7I ' J ^ (i 4-*) 2 ' 

f ^r 
72- )x(a + bxy 

73 ' J * 4 (a 4- A* 8 ) ' 



log 


X 

— i — + 

I + .* 


I 

I + X° 




— log- 

32 to « 


X* 

+ bx ?s 


— ^+ 

3^^ 




+ bx* 

x s '■ 



74. Find the whole area enclosed by both loops of the curve 

/ = *»(!-.*•). 4. 

75. Find the area enclosed between the asymptote corresponding 
to x = a, and the curve 

x*y* + a*x* = a 2 y 2 . 20 s '. 

76. Find the whole area enclosed by the curve 

a*/ = x" (a' - x*). \a % 

77. Find the area enclosed by the catenary 
the axes and any ordinate. 



fr*--*} 



78. Find the whole area between the witch 
xy* = 4a 1 (2a — x) 
and its asymptote. See Ex. 23. 



4x0*. 



§ III.] 



TRIG 0X0 ME TRIC IN TE GRA L S. 



33 



III. 



Trigonometric Integrals. 

25. The transformation, tan 2 = sec 2 6 — I, suffices to 
separate all integrals of the form 



tan»0</6>, (i) 



in which n is an integer, into directly integrable parts. Thus, 
for example, 



tan 5 6 dd 



tan 3 6 (sec 2 d - i) dd 

tan 4 6 [ , . , Q 
tan 3 S dd. 



Transforming the last integral in like manner, we have 

f - Q ._ tan 4 6 tan 2 6 f „ 7/1 
tan 5 0//# = + tan 0<#; 

hence (see Art. 12) 



L 5/1^ tan4 # tan2 # 1 z, 

tan 5 Odd = log cos 6. 

J 4. 2 

When the value of n in (1) is even, the value of the final inte- 
gral will be 6. When n is negative, the integral takes the form 

[cot* 6 dd, 
which may be treated in a similar manner. 



34 ELEMENTARY METHODS OF INTEGRATION. [Art. 2& 

26. Integrals of the form 

[sec n 6dd (2) 

are readily evaluated when n is an even number ; thus 
[sec 6 ddO = [(tan 2 d + i) 2 sec 2 Odd 

= [tan 4 sec 2 dd + 2 [tan 2 sec 2 0*0 + [sec 2 </0 

tan 5 2 tan 3 

= H ■ + tan 0. 

5 3 

If n in expression (2) is odd, the method to be explained in 
Section VI is required. 

Integrals of the form cosec w 6 dd are treated in like manner. 

Cases in which sin™ cos* dd is directly integrable. 

27. If n is a positive odd number, an integral of the form 

[sin w cos" 6 dS ....... (3) 

is directly integrable in terms of sin 0. Thus, 

Jsin 2 cos 5 0^0 = [sin 2 (1 - sin 2 0) 2 cos Odd 
_ sin 3 2 sin 5 sin 7 

This method is evidently applicable even when m is frac- 
tional or negative. Thus, putting y for sin 0, 



§ III.] TRIGONOMETRIC INTEGRALS. 35 

hence 

f cosB 6 jn , l 2 » 2 3 + sin 2 

When ^ in expression (3) is a positive odd number, the in- 
tegral is evaluated in a similar manner. 

28. An integral of the form (3) is also directly integrable 
when m + n is an even negative integer, in other words, when it 
can be written in the form 



'sin w 6 dB 
cos w+2 ? d 



[tan™ 6 sec 2 ? 6 dd, 



in which q is positive. 
For example, 

f d6 



sin* 6 cosa 



f (tan 0)~i sec 4 6 dd 
(tantf)-t (tan 2 # + 1) sec*ed8; 



hence 

f dd 2 a 2 

. a 4 5— = 7 tarn - - — — - 

J sins cos 2 3 tan 2 6 

It may be more convenient to express the integral in terms 
of cot and cosec 6, thus 

•cos^ 



. 8 ^ = cot 4 (9 (cot 2 0+i) cosec 2 
cot 7 cot 5 



36 ELEMENTARY METHODS OF INTEGRATION. [Art. 28. 

Integrals of the forms treated in Art. 25 and Art. 26 are in- 
cluded in the general form (3), Art. 27. Except in the cases 
already considered, and in the special cases given below, the 
method of reduction given in Section VI is required in the 
evaluation of integrals of this form. 



The Integrals sin 2 6 d6, and cos 2 Odd. 

29. These integrals are readily evaluated by means of the 

transformations 

sin 2 8 = -J-(i — cos 28), and cos 2 6 = !(i -f cos 28). 
Thus 

ism 2 8 d6 - \\d6 - J [cos 28 d8 = \8 - isin 2(9, 

or, since sin 26 — 2 sin 6 cos 6, 

[sin 2 6d8 = 4(6> - sin 6 cos S) (B) 

In like manner 

[ cos 2 8 d6 = i (8 + sin0cos6>) (Q 

Since sin 2 (9 + cos 2 6 = 1, the sum of these integrals is L/0; ac- 
cordingly we find the sum of their values to be 6. 

In the applications of the Integral Calculus, these integrals 
frequently occur with the limits o and \n ; from (B) and (C) 
we derive 



[ 2 sin 2 ^6> = [ 2 cos 2 0^ = i;r. 



§111.] TRIGONOMETRIC INTEGRALS. 37 



The Integrals — , — — - , and . 

s J sin 6 cos 6 J sin 6 Scosd 

30. We have 

f dd [sec 2 Odd . 

-r— a 2 = — — jj- = log tan 0. . . . (Z> 

J sin 6 cos 6 J tan & v ; 

Again, using the transformation, 

sin 6 = 2 sin |# cos £#, 

we have 

[dd_ _ f f^9 t sec* \B \d6 1 

Jsin^^Jsin^cosi^ - ] tan \6 ' 

hence 

J^=logUntf 0) 

This integral may also be evaluated thus, 
dd f sin Odd f smddd 



f dffl _f sin flaffl _f sin 
JsIn0~~J sin 2 ^J i - 



cos* 



Since sin Odd = — ^(cos 6*), the value of the last integral is, by 
formula^'), Art. 17, 

1 , 1 — cos /I — cos 



j. . 1 — cos V _ . / I — 

2 ° g I + cos ~~ ° g V I + 



costf' 



and, multiplying both terms of the fraction by 1 — cos 6, we 
have 

f dd , 1 — cos 8 , ~, N 



38 ELEMENTARY METHODS OF INTEGRATION. [Art. 3 1. 

31. Since cos 8 = sin (%n 4- 0), we derive from formula (E), 

f dS f dd . . Vn 8~] /Z7 , 

]^o^=]sin-(R^ j = 10gtan L4 + 2j- ' ■ {F) 

By employing a process similar to that used in deriving for- 
mula (£'), we have also 

f dd - ■ 1 + sin , _.,. 



cos & cos 



Miscellaneous Trigonometric Integrals. 

32. A trigonometric integral may sometimes be reduced, 
by means of the formulas for trigonometric transformation, to 
one of the forms integrated in the preceding articles. For 
example, let us take the integral 

f dd 



)a sin 8 -+- b cos u 

Putting a = k cos a, b = k sin a, . . . ._ . - (1) 

we have 

f dd £ f ^ 

J sin + £ cos 8 ~ k J sin (8 + a) * 

Hence by formula (£) 

== - log tan - (8 + a) ; 

Jtfsin<9+ £cos<9 £ 5 2 V " 

or, since equations (1) give 

k= V(a* + P), tana = -, 

J sin 6* + * cos 8 \/{a 2 + b 2 ) s 2 |_ « J 



§ III.] MISCELLANEOUS TRIGONOMETRIC INTEGRALS. 39 

33. The expression sin md sin nddd maybe integrated by 
means of the formula 

cos (m — n) — cos (m + n) 6 = 2 sin md sin nd ; 
whence 

f . - , a sin On -?i) 6 sin (m + n)d 

sin md sin nd dd = - — -. — j — ~ . , (i) 

J 2 {in — n) 2 {in + n) 

In like manner, from 

cos {m — n) 6 + cos (m + n) — 2 cos md cos nd, 

we derive 

f „ „ Jn sin (/zz — «) sin (/« + n) d . . 

\cos md cos nddd = } f- + } (— . . (2) 

J 2 {111 — «) 2 (in + n) 

When m = n, the first term of the second member of each 

of these equations takes an indeterminate form. Evaluating 

this term, we have 

f • 9 n jn Q sin2ii8 , x 

sin 2 /z0^0 = , . . . . (3) 

J 2 47* ' VJy 



sin 8 «0</0 = -- 


sin z.nv 


4n 


cos 2 nddd = - + 


sin 2?z0 



and \cos 2 nddd = ~ + " (4) 

J 2 4^ 

Using the limits o and 7r we have, from (1) and (2), when m 
and n are unequal integers, 

sin md sin nd dd -\ cos md cos nd dd = o ; . . (5) 

J o 

but, when m and /z are equal integers, we have from (3) and (4) 
[ sir? nOdd = fcos 2 ^^ = - (6) 

Jo Jo ^ 

34. To integrate V{i + cos 0) </0, we use the formula 
2 cos 2 £0=i+ cos 0. 



40 ELEMENTARY METHODS OF INTEGRATION. [Art. 34. 

whence 4/(1 + cos 8) = ± s/2 cos £0, 

in which the positive sign is to be taken, provided the value oi 
Q is between o and zr. Supposing this to be the case, we have 

J V (1 + cos 6) dd = 4/2 [cos \kdQ 

= 24/2 sin £0. 
For example, we have the definite integral 

i 2 V{i + cosd)d8 ^ 2 V2 sin 7=2, 

Jo 



4 



Integration of j-. Ti . 

s J a + b cos 

35. By means of the formulas 

1 = cos 2 \e + sin 2 \d and cos 6 = cos 2 J0 — sin 2 \6, 
we have 

Ja + £cos ~ J (a + b) cos 2 i# + (a - b) sin 2 ^* 

Multiplying numerator and denominator by sec 2 |#, this be« 
comes 

f sec 2 jOdd 

and, putting for abbreviation 

tan \6 — y, 
we have, since \ sec 2 \6dB = dy, 

Ja + bcosd ~ 2 }a + b + (a- b)f 



§ III.] MISCELLANEOUS TRIGONOMETRIC INTEGRALS. 41 

The form of this integral depends upon the relative values 
of a and b. Assuming a to be positive, if b, which may be 
either positive or negative, is numerically less than a, we may 
put 

a ± & — 2, 
a — b 

The integral may then be written in the form 

dy 



a — b 



*+f % 



the value of which is, by formula (k') f 



tan" 1 ^. 



c(a-b) 



Hence, substituting their values for y and c, we have, in this 
case, 

l a -ffcos = V (/- *■) **"' '[VjtI** 1 **} • (G) 

If, on the other hand, b is numerically greater than a, this 
expression for the integral involves imaginary quantities; but 
putting 

^ + a _ a 

the integral becomes 

2 f dy 

the value of which is, by formula (A f ), Art. 17, 

1 log' + ' 



c(b — a) c — y 



42 ELEMENTARY METHODS OF INTEGRATION. [Art. 35. 



Therefore, in this case. 



t 



dS 



i log V{b + a)+V(b-a)t*xi\0 ^ _ 



+ £ cos V (£ 2 - a 2 ) 8 •(£ + *)- y(£-*) tan J 
36. If * < 1, formula (£) of the preceding article gives 
dd ■?. . r /t - p 1 



cos 6* 4/(1 — ^) 



tan 



1 / rT - / an^J. .(,) 



Putting 



|/f T ^-tani(9 = taniA (2) 

and noticing that $ = o when 6 = o, we may write 



J.TT 



e cos |/ (1 — e 2 ) 



(3) 



Now, if in equation (1) we put <j> for 6 and change the sign of 
e, we obtain 



1 — ^cos (/> V{i — e 1 ) 

hence, by equation (2), 

d</> 



tan 



- 1 



Vrz- tan * A 



\s= 



e 



and 



e cos (/> V (1 — e 1 ) 

Equations (3) and (4) are equivalent to 

dd _ d(f> 
I + e cos6~ 4/(1 -e*y 

dS 



1 — ^cos (f> V (1 — e 2 ) ' 



(4) 

(5) 

(6) 



§ III.] TRIGONOMETRIC INTEGRALS. 43 

the product of which gives 

(i + e cos 6) (i — e cos ft) = i — & , , . (f\ 

By means of these relations any expression of the form 

f </fl 

J (i + <? cos#)*' 

where n is a positive integer, may be reduced to an integrabie 
form. For 

f dB f ^ I . 

J(i + * cos#)« ~ J i + e cos 6 (i + e costf)*" 1 

hence, by equations (5) and (7), 

I 7 : 7K7, = r jl\„ x I (* ~ e COsft) n - x d(t). 

J o (i + ezosvy (i-^) w -*J v 

By expanding (1 — e cos ft) n ~ x , the last expression is reduced 
to a series of integrals involving powers of cos & ; these may 
be evaluated by the methods given in this section and Section 
VI, and the results expressed in terms of 6 by means of equa- 
tion (2) or of equation (7). 



:. \tan*mxdx, 



Examples III. 

tan 3 mx tan mx 

— + x. 

$m m 



2. * tdJi'xdx, 1^ — ilog2. 

Jo 

(« /„ \ 7 tan 3 (6 + a) fn , 

sec 4 (0 + a) d% * - + tan (9 + a) 

o 



44 ELEMENTARY METHODS OF INTEGRATION. [Ex. III. 

V A 

4. sin mxdx, $m 

Jo 

5. sin 8 6 cos 3 dQ, 

6. |/(sin 0) cos 5 </0, 

7. 2 cos 4 sin 3 dfe, 



2 . i 4 . L 2 . ii. 

- sin*0 sin 8 -\ sin 2 

3 7 11 



2 

35 



f sin 3 de & L 

1 Jy(cos0)' f cos* - 2 cos 8 0. 

f dB 

>• ^"^ rr > Multiply by sin 2 + cos 2 0. tan — cot 0. 

J sin cos 



(sin 8 * , A . tan x 

10. — r— </#, See Art. 28. 

Jcc- 5 



cos j; 



11 



. I . , ° 3 , | (tan 2 - cot 2 0) + 2 log tan 0. 

J sin 3 cos 3 9 ' * 

fi/(sin0)^0 „ 3 

J cos 2 



f si 

I3 - J- 



sin # afo: 



cos r '* ' 5 cos 5 * 3COS 3 X* 



14 



tan # tan x 
5 3 



(sin 2 # tfk 
COS je 

. I sin 3 cos 2 */0, lV [20 — sin 20 cos 20] 



§ in-: 



EXAMPLES, 



45 



16 



17 



IT 

[m . 2 

sm mx ax, 

•'o 

f sin 2 afc 
J cose ' 

n 

■l; 



2W 



2 cos 



Z sin 1 

3 



f </0 
9 * J sin + cose' 

-In? 



COS .# 

dx 



21, 



22. 



COS X' 



\~7Z e~] 
log tan — I — — sin e. 

i(log3 -1). 

tan Jjc. 
1 — cot J*. 



fr 



*fo 



± sin x f 
Multiply both terms of the fraction by 1 =f sin #. tan x =F sec .r. 



4 



'sec 9 ± tan0' 

24. cos cos 36 dd. See Art, 33. 

it 

25. 1 COS COS 2QdB 9 



log tan — 



± log cos 0. 



7T 

26. 4 sin 2 si 

Jo 



sin 20 dB y 



it 

-f 



27. sin 30 sin 20 <#, 



i sin 40 + J sin 20. 

1 
3° 

isin 4 0l 4 =i 

— 'o 

2 
5' 



46 ELEMENTARY METHODS OF INTEGRATION. [Ex. III. 



28. sin mO cos nti 



1 — cos (m + n) B 1 — cos (m — n) 
2 (m -\- n) 2 (m — n) 



29. cos x cos 2x cos $x dx, 



Reduce products to sums by means of equation (2), Art. ^. 

1 Tsin 6x sin 4X sin 2# 



4 L 6 



o. [ V (1 — cosx)dx, 

J; 



+ * . 



2 ^2, 



31 



32. 



33- 



s 


2 

COS 

dx 


# + 


b 2 sin 2 * ' 


I 


+ cos 2 .* ' 








dx 




$ 


cos 2 


x — 


£ 2 sm 2 *' 






sin x dx 



1 xVb 1 

— tan — tan* . 
00 \_a J 



1 t tan# 

— - tan" 1 — — 

4/2 |/2 



log 



a + b tan x 



2 o$ a — b tan a: 



J V (3 cos 2 .# + 4 sin 2 #) 

f sin # cos 2 # dx 
35 ' J 1 + a' cos 2 * ' 

f y*dy 
Putting y /<?/- cos x, the integral becomes — — — — 5 



cos * {-J- cosjc}. 



cos # tan * (# cos #) 

i_ - '- 

a ' 3 



§ III.] 



EXAMPLES. 



47 



J a 4- b sin 9 

/>«/ sin 9 = cos (<j — \7t), and use formulas (G) and (G'). 

/a — b 25 — 7T~] 

J 7 tan . 

y a 4- b 4 J 



If a > b, 



tan" 1 



If a< 



V (a 2 - P) 

V (b 4- a) + V (b - a) tan (J 9 - j ?r) 



1/ (£ 2 - flf 2 ) l0g V{b + a)- V {b-a) tan (J 5 - 



de 



37- 



*b 



3 + 5 cos 5 



ilog 



2 4- tan I 



5 4- 3 cos 



39- 



40. 



f a\ 

J5 -4 

J 



cos 



2 COS 6 — I 



41 



42 



■ f" — 

Jo 3 - 

71 

•f 



3 — cos 
dB 



2 — tan i 9 
i-tan-^l-tanio]. 



I tan" 1 -! 3 tanl 0J. 



1 1 4- 4/3 tan 1 

V3 ° g i — V^Tan-J-? 

tan" 1 V2 

V2 



2 — cos B 



2 V3 



43- 



W 



e cos 6) 



, .&tf Art. 36. 



cos 



, ^ 4- cos 9 



sm 5 



( x _ e *\i 1 4- ^ cos 9 1 — er 1 4- <? cos S 



( n dS 

44< J (1 + *cos9) 3 ' 



(2 4- f) n 
2(1- * 2 )* 



48 ELEMENTARY METHODS OF INTEGRATION. [Ex, III,. 



f p cos x + qsmx . 

ak. - *r-. — dx, 

J acosx + osmx 



Solution : — 

By adding and subtracting an undetermined constant, the fraction 
may be written in the form 

p cos x + q sin x + A (a cos x 4- b sin x) 

~ r^~^ - A > 

a cos x + b sin x 

we may now assume 

/ cos x + q sinx + A (a cos x + b sin x) = k (b cos x — a sin x) ; 

the expression is then readily integrated, and A and £ so determined 
as to make the equation last written an identity. The result is 

f p cos x + q sin x , ap + bq bp — aq , . » 

J a cos # + b sin * a 2 -M a 3 + b 2 b v ' , 

46. — — tt > See Ex. 45. 

J a + £ tan ar 

ax , b . , , . v 

+ -g 72 log (a cos # 4- b sm a:). 



** 4- b* a* + b* 

47. Find the area of the ellipse 

x = a cos $ y = £ sin ^. 



— 40^ sin 2 */$ = 7ra£. 



48. Find the area of the cycloid 

x = a (ip — sin tp) y = a (1 — cos ^). 



(27T 
(1 - cos ipY dip = 3 a'7r, 



§111.] EXAMPLES. 49 

49. Find the area of the trochoid (b < a) 

x = aip — b sin ip y = a — b cos ip. 

(2a 2 + b") n. 

50. Find the area of the loop, and also the area between the curve 
and the asymptote, in the case of the strophoid whose polar equation is 

r = a (sec ± tan S). 
Solution : — 
Using as an auxiliary variable, we have 

sin 2 0~1 
x — a (1 ± sin o) y = a tan ± , 

L cos 0_r 

the upper sign corresponding to the infinite branch, and the lower to 
the loop. Hence, for the half areas we obtain 

sin 6de + o?\ sin 2 dQ = a 2 1 + - 

and — a 2 \ sin d% + a* f sin 2 dQ = tf 2 1 . 

J*- Ji. L 4J 



50 METHODS OF INTEGRATION. [Art. 37. 

CHAPTER II. 

Methods of Integration — Continued. 



IV. 

Integration by Change of Independent Variable. 

37. If x is the independent variable used in expressing an 
integral, and y is any function of x, the integral may be ex- 
pressed in terms of y, by substituting for x and dx their values 
in terms of y and dy. By properly assuming the function y, 
the integral may frequently be made to take a directly integra- 
te form. For example, the integral 

x dx 



(ax + bf 
will obviously be simplified by assuming 

y = ax + b 
for the new independent variable. This assumption gives 



x = - , whence dx = — ; 

a a 


substituting, we have 


f x dx 1 
J (ax + bf ~ a 2 . 


H y - b) dy 


= a* 


lo ^ + ^; 



§ IV.] CHANGE OF INDEPENDENT VARIABLE. 5 1 

or replacing y by x in the result, 

(x dx i . , 1X 



(ax + bj a* s y } a?(ax + b)' 

38. Again, if in the integral 

f dx 
)e — i 

we put y = e% whence 

dy 
x = log y, and dx = — , 

y 

we have 

dx _ [ dy 



H 



J/(j- I) 

Hence, by formula (A), Art. 17, 

\e^r Y = lo % y -jr = l °s {* ~ - *• 

It is easily seen that, by this change of independent variable, 
any integral in which the coefficient of dx is a rational func- 
tion of £ x , may be transformed into one in which the coefficient 
of dy is a rational function of/. 



Transformation of Trigonometric Forms. 

39. When in a trigonometric integral the coefficient of dd is 
a rational function of tan 6, the integral will take a rational 
algebraic form if we put 

tan 6 = x, whence dd = — - — 3- . 

1 + x^ 



52 METHODS OF INTEGRATION. [Art. 39. 

For example, by this transformation, we have 

f dd _ f dx 

J I 4- tan 6~ J (1 +^)(i + *)' 

Decomposing the fraction in the latter integral, we have 

f dd _ \[ dx _ I f ,r^r ^il dx 

J 1 + tan 6* ~~ 2J 1 + ^ 2 jT+7 ' 2)1 + x 

= i tan -1 * - i log (1 + ,r 2 ) + J log(i + x) 

1 + tan d~ 



6 + log 



sec 



l + tan ^ = i[0 + lo g ( cos * + sin #)]• 



40. The method given in the preceding article may be em- 
ployed when the coefficient of dd is a homogeneous rational func- 
tion of sin 6 and cos 9, of a degree indicated by an even integer ; 
for such a function is a rational function of tan d. It may also 
be noticed that, when the coefficient of dd is any rational func- 
tion of sin 6 and cos d, the integral becomes rational and alge- 
braic if we put 

d 



tan 



for this gives 



• n 2Z n l — £ jn 2 d% 

sin d — 9 , cos d = -o , dd = 



This transformation has in fact been already employed in 

the integration of ■ - . See Art. 35. 

& a + b cos 6 DD 



§ IV.] LIMITS OF THE TRANSFORMED INTEGRAL. 53 



Lzjnits of the Transformed Integral. 

41. When a definite integral is transformed by a change of 
independent variable, it is necessary to make a corresponding 
change in the limits. If, for example, in the integral 

r dx 

J„(a» + x 2 ) 2 
we put x = a tan 0, whence dx = a sec 2 Odd, 

we must at the same time replace the limits a and oo , which 
are values of x, by \n and \n, the corresponding values of 6. 
Thus 

w 

dx 



f _*£ JLf'cos.^ 

i a (a l + x 2 y a s }n_ 



~ 2a\_ 



v + sin u cos I = — — - 

J.ir 8tf 



77z£ Reciprocal of x taken as the New Independent 

Variable. 

42. In the case of fractional integrals, it is sometimes use- 
ful to take the reciprocal of x as the new independent variable. 
For example, let the given integral be 

f dx 



\x\x + i) 2 * 

i dy 

Putting x=~, whence dx = 5-, 

O y yi 1 



54 METHODS OF INTEGRATION. [Art. 42. 



we have 



dx _ f f dy _ f fdy 

J^(^+i) 2 -"J ./ , iV-~J(37Tl 



Transforming again by putting z — y + I, the integral be- 
comes 



= - ~ + 3* - 3 log z - - 



Therefore, since z =y + 1 = - + 1 = , 

x x 



dx I 2 I # + I 

= - ^T2 + ~ + r-T-T-3.log-— +C. 



J (# + i) 2 2* 2 x x -f 1 



43. In the above example we see that the single substitu- 

x -f- 1 
tion # = is equivalent to the two substitutions which 

are suggested in the process. In like manner, in the case of the 
more general integral 

f dx 



\{x —d) m {x— bf } 

the successive transformations which suggest themselves are 
found to be equivalent to a single one in which the new vari- 
able is 

x — a bz — a 

z = 7, whence x = — . 

x — b z — 1 



§ IV.] TRANSFORMATION TO A POWER OF X. 55 



The result is 

I f(I — z) 1 



-dz, 



which, when m + n — 2 is a positive integer, is directly inte- 
grate after expansion by the Binomial Theorem. 



A Power of x taken as the New Independent Variable. 
44. The transformation of an integral by the assumption 

y = * n (j) 

is not generally useful, since the substitution 

I L_, 

x = /\ whence dx = -y n dy, 

will usually introduce radicals. Exceptional cases, however, 
occur. For, since logarithmic differentiation of equation (i) 
gives 

dx dy 

— = — , (2) 

x ny J 

it is evident that, if the expression to be integrated is the product 
of — and a function of x n , the transformed expression will be 

dy 

the product of — and the like function of y. 

r ny 

For example, the substitution y = x* transforms 

(x* — \)dx . (y — \)dy 
i-r-i — '-r- into —, ff. 

x(x* + 1) Ay{y + 1) 



56 METHODS OF INTEGRATION. [Art. 44. 

Hence, decomposing the fraction in the latter expression, 
f (* - \)dx 1 (y + i) 2 +/Q 4 + I) 

T r- = _ lOff = lOg . 

J x(x 4 + 1) 4 J * 

Again, putting ;tr 3 = y 2 or jy = # ? , we have 

Jdx 2[ dy 2 _ x y 2 ~! [ x \^ 

x \/(x z -a*) ~ 3J y V{f—a 3 ) ~ ^f Se ° «f ~ ^i Se ° W' 

4-5. When this method is applied to an integral whose form 
at the same time suggests the employment of the reciprocal, 
as in Art. 42, we may at once assume y = x' n . Thus, given 
the integral 

Idx 
x^(2+^) ; 

putting y — x~ 3 , whence — = — • -2 

we obtain 

dy 



[" dx _ 1 f° yd} 

J x x 4 (2 + X s ) ~~ 3 J s 2y 4- 1 

J , log (27 + 1) 1° __ 2~ log 3 
6 12 J, 12 

Examples IV. 



§ IV.] EXAMPLES. 57 

X dx 2X — I 



k. 



x) 3i 2 ( I _ JC )»- 



2X + I log \2X + i) 



6 



3 * J (2^+ l) 2 ' 8 2 8(2* + l)* 

f ^Vi 4y — 2~| 2 I 



• Jtti^ *-io g (i + £-). 

f <&■ I £■* — I 



7 * Ll^TT' i -log 2. 

8 - f /l + g ^ ^> ** + 2 lo s ( £ * - r ). 

f 2 + tan Q 6 — log (3 cos — sin 0) 

Q. 1 dQ» — — — ■ ■. 

y J 3 — tan ' 2 

f ^ T 1 

o. 7, . — lOff 

j tan' - 1 ' 4 to 



1 , tan 0—1 



4 tan 0+1 2 



f tan d0 1 . tan — 1 , 6 

11. — s , - log +- 

J tan — 1 ' 4 & tan + 12 

f cos dQ aB — b log (a cos — <5 sin 0) 

J a cos — b sin ' a* + b 2 



58 



METHODS OF INTEGRATION. [Ex. IV. 



13- 



f cos e de 

J cos (a + e) 1 



Put Q' = a + 0. 

(0 + a) cos « — sin « log cos (0 + a). 



14 



[ sin (s + < ,, 





(0 + /?) cos (a—/3) + sin (a — /3) log sin (6 + /?). 



5. tan (0 + «) cos */0, —cos + sin a log tan 



20 + 2« + 7T 



f a COS */0 

Jo sin (a 4- 0)' 



cos a log (2 cos a) + a sin a. 



1: 



3 cos I . I - ^2 +2S1I1( 
17. I —d6, —7- log— 7 r— , 

cos 4/2 ° 4/2 — 2 sm 



6 _ log( 3 + 24/2; 

o V2 



,f 



sin -J dQ 
sin0 ' 



log tan 



7t 4- 



19- 



f x* dx 



ilog(^ + ^ 2 ) + 



2 (a 2 4- ^ 2 ) 



20 ' j^ 3 (i + -*V 



log 



|/(l + x 2 ) I_ 

X 2X l 



21 



' Jc (1 + *T 



r 

4Jc 



sin 8 2B dd= — t- 
16 



22. 



f 00 dk 



I — log 2 



§ IV.] EXAMPLES. 59 

(dx I , I , x + I 

F^+Tj' -^ + " ' log 



2X X X 



X 



I — X 



2 - log 3 



r <&• i i M 

4 ' J(i-^) 3 ^' 2 (i-xy + i-x^ 10g 

* : &^W -^ + iiog( 2 / +I) ];= 

6 ' J x(a + bxy Ja g a + dx* 

r dx i 



#0" °.r A + a 



X" 

n' 



f(^ 3 + l)dX 9, / 3 \ 1 

28 ' J *(*»-i) ' |log(x 3 -i)-log^ 

n 

(dx 2 /x x 2 



V. 

Integrals Containing Radicals. 

46. An integral containing a single radical, in which the 
expression under the radical sign is of the first degree, is 
rationalized, that is, transformed into a rational integral, by- 
taking the radical as the value of the new independent vari- 
able. Thus, given the integral 

f dx 



i + V(* + i) ' 



6b 



METHODS OF INTEGRATION. 



[Art. 46. 



putting 
whence 
we have 



y = tf( x + I), 

x — f — i, and 



dx = 2jJ/ ^, 



f dx _ 

J I + V(^r + I) " 



y dy 

1 +j/ 



^- 2 (- 



^/ 



+ .7 

= 2J- 2 log (I +7) 

= 2|/(JT+I)- 2 log [I + ^ + I)]. 



47. The same method evidently applies whenever all the 
radicals which occur in the integral are powers of a single 
radical, in which the expression under the radical sign is linear. 
Thus, in the integral 



dx 



,(,_ : i)I + (*_!) 



1 » 



the radicals are powers of (x — i)£ ; hence we put y = (x — i)£, 
and obtain 



dx 



z {x-i)* + {x- 1) 



1 = 6 



= 6 



1 y ^ 

c /+7 3 

j>- 1) ^ + ^7+1" = -3 + 6 lo S 2 - 



48. An integral in which a binomial expression occurs 
under the radical sign can sometimes be reduced to the form 
considered above by the method of Art. 44. For example, 
since 

f dx 
ixix 3 + i)i 



§ V.] INTEGRALS CONTAINING RADICALS. 6l 

fulfils the condition given in Art. 44 when n — 3, the quantity 
under the radical sign may be reduced to the first degree. 
Hence, in accordance with Art. 46, we may take the radical as 
the value of the new independent variable. Thus, putting 



whence x* = z 4 —i, and — = —$-. c, 

x 3(^-1) 

we have 



dx _ 4 
x{x* + if~ 3 



\£dz 



jl 



Decomposing the fraction in the latter integral (see Art. 22), 
we have finally 

[ dx 2 . T /Jt ■ill, (**+ l)*-I 
-vr = - tan \ (x 3 + 1) N — log- \ , 



Radicals of the Form ^(ax 1 + $). 

49. It is evident that the method given in the preceding 
article is applicable to all integrals of the general form 

|:r Jw+I (tf^ 2 + b)*+*dx> . . . . . (1) 

in which m and n are positive or negative integers. These 
integrals are therefore rationalized by putting 

y = ^(ax 2 + b). 



62 METHODS OF INTEGRATION. [Art. 49, 



Putting m = O, the form (1) includes the directly integrable 

I (ax 2 + b) n + * xdx. 

50. As an illustration let us take the integral 

f dx 

J x V(* 2 + a 2 ) ' 



dx _ jk^ 



putting 7 = V(x* + # 2 ), 

whence x 2 =y i — a 2 , and 

we have 

f dx _[ dy 

}x~Y(x 2 T~a T )~\f-d i ' 

Hence, by equation (A') Art. 17, 



<^t: _ I . y ~ a _ l \ ^{ x<i + d 1 ) — a 

x V(x 2 + a 2 ) ~2a° S y + a ~ ~2a ° g ^(^ +~# 2 ) + # 



Rationalizing the denominator of the fraction in this result, 
we have 

V{x> + a 2 ) -a [ V(x 2 4- a 2 ) - af 



Vix 2 + a 2 ) +a x 2 

Therefore 

f dx 1 , V(x* + a 2 ) — a , „. 

]x^(x 2 + a 2 ) a & * v ' 



§ V.] INTEGRALS CONTAINING RADICALS. 63 
In a similar manner we may prove that 

f dx 1 . a — V(a 2 — X s ) /T . 

— 77-2 — ^- = - l °g — '• • • • CO 

J x V(a z — x 2 ) a x v ' 
51. Integrals of the form 

\&~(aa? + b)*+*dx (2) 



are reducible to the form (1) Art. 49, by first putting y = - , 

x 

For example : 

r dx 

\(ax* + bf 



is of the form (2) ; but, putting x — - , whence 



V(ax* + t)= ^ a+ W and dx = - % % 

y f 

we obtain 

dx I* y dy 



[ dx = _ r j/ 

The resulting expression is in this case directly integrable. 
Thus 

f dx _ * . _ * (/\ 



64 METHODS OF INTEGRATION. [Art. 52. 



Integration of 



V(x* ± a 2 ) ' 

52. If we assume a new variable z connected with x by the 
relation 

z — x= V{**±a 2 ), (1) 

we have, by squaring, 

z* — 2zx = ± a 2 , (2) 

and, by differentiating this equation, 

2 (z — x) dz — 2z dx = o ; 
whence 

dx dz 



z — X z 
or by equation (1), 

dx dz 



■ (3) 



V{x* ±a 2 )~ z 

Integrating equation (3), we obtain 

1 V(f± * ) = '°g^ = lQ g \* + V{*± ^)] ■ • • (K) 

53. Since the value of x in terms of z, derived from equa- 
tion (2) of the preceding article, is rational, it is obvious that 
this transformation may be employed to rationalize any ex- 

dx 
pression which consists of the product of— ,- 2 ^ and a 

rational function of x. For example, let us find the value of 

\v{x*±a i )dx ) 



§ V.] TRIGONOMETRIC TRANSFORMATION. 6$ 

which may be written in the form 

dx 



JV ± * 2 ) 



By equation (2) 



# =F a 2 
whence 

"*'-<*£* (5» 

Therefore, by equations (3) and (5), 

4J 2 J 2* 4 J 2 s 

a 4 - tf 4 « 2 . 

= S~9 ± — 1°& #• 

By equations (4) and (5), the first term of the last member 
is equal to J x V{x* ± a 1 ). Hence 

J •(*» ± * 2 ) dx = xV ( x *± a2 ) ± a l i og [> + ^ ± ^)j . . (Z ) 

Transformation to Trigonometric Forms. 
54. Integrals involving either of the radicals 
•(a 2 -* 8 ), •(* a + * a )> or V(**-d) 



65 METHODS OF INTEGRATION. [Art. 54, 

can be transformed into rational trigonometric integrals. The 
transformation is effected in the first case by putting 

x = a sin 8, whence \/(a 2 — x 2 ) = a cos 6 ; 

in the second case, by putting 

x — a tan 6, whence V(a 2 + x 2 ) = a sec 6 ; 

and in the third case, by putting 

x = a sec 0, whence tf(x* — a 2 ) = a tan 0. 

55. As an illustration, let us take the integral 

f V(a 2 - x 2 ) dx ; 

putting x = asm. 0, we have V(a 2 — x 2 ) — a cos d,dx = a cos 6 dd; 
hence 



^(tf 2 — x 2 ) dx — a 2 \ cos 2 6 



d6 



by formula (C) Art. 29. Replacing 6> by „r in the result, 

Regarding the radical as a positive quantity, the value 

of may be restricted to the primary value of the symbol 

x 
sin -1 — (see Diff. Calc, Art. 54) ; that is, as x passes from — a 

to + a, 6 passes from — £ n to + \ 71. 



§V.] INTEGRALS CONTAINING RADICALS. 67 



Radicals of the Form ^/{ax 2 + bx + c). 

56. When a radical of the form Viax 2 + bx + c) occurs in an 
integral, a simple change of independent variable will cause the 
radical to assume one of the forms considered in the preceding 
articles. Thus, if the coefficient of x* is positive, 

•(«*»+ bx + c) = Va |/[(* + £j + 4 ^7 ^ ] , 

in which, if we put x + — = y, the radical takes the form 

Vif* + a 2 ) or V(y* — # 2 ), according as ^ac — b 2 is positive or 
negative. If a is negative, the radical can in like manner be 
reduced to the form V(a 2 — y 1 ) or V(— a 2 — y~) ; but the latter will 
never occur, since it is imaginary for all values of y, and there- 
fore imaginary for all values of x. 

For example, by this transformation, the integral 

r dx 



J (ax 2 + bx + c)'~ 
can be reduced at once to the form (y), Art. 51. Thus 
dx f dx 



I ax e ax 



4ac — b 2 
4a 



2a ^ax + 2b 



- b \ { a^ + bx + c)~^ C -^^ + bX + C) 



4a 



68 METHODS OF INTEGRATION. [Art. $J t 

57. When the form of the integral suggests a further 
change of independent variable, we may at once assume the 
expression for the new variable in the required form. For 
example, given the integral 

V(2ax — x 2 ) x dx ; 

we have V(2ax — x 2 ) = Via 2 — (x — a) 2 ] 

hence (see Art. 54), if we put x — a — a sin 6 y we have 
V{2ax —x 2 ) = a cos 6, 
x = a ( 1 + sin 6), dx = a cos 6 dd ; 

.-. V(2ax — x*)xdx = a 2, cos 2 6(1 + sin 6) dd 

= — (6 + sin 6 cos 6) - — cos 3 B 
2 3 

tf 3 . .r — a a , s „ 9X 1 , _ s 

= — sin" 1 + —{x — a) V(2ax — x 2 ) — - (2ax - x 2 )* 

2 a 2 v v 3 



2 



sin - x h -p- V(2ax — x 2 ) [2x* — ax — 3a 2 ]. 



The Integrals 
dx , r dx 



w\(x-«Mx-es\ and J 



v[(* - «)(* - /*)] J vK* - «)(/* - *)] ' 

58. An integral of the form — -, — may by the 

method of Art. 56, be reduced to the form (K), Art. 52, or to 
the form (/'), Art. 10, according as # is positive or negative. 



§ V.] IRRATIONAL INTEGRALS. 69 

But, when the integral appears in one of the above forms (the 
quadratic under the radical sign admitting of linear factors), 
another mode of transforming is often convenient. 

Assuming /? > a, we may in the first case, since the differ- 
ence of the factors is the constant (3 — a, put 



whence 
and 



x — a = (/? — a) sec 2 ) 
x - ft = (/? - a) tan 2 6 \ 



dx = 2(J3 - a) sec 2 6 tan 6d6, 



dx 



2 J sec dO = 2 log (sec 6 + tan 6). 



j V [_( X - a)(x - /i)] 
Substituting, and omitting the constant log \\ft — ct), 

Jdx 
A{x _ a){x _ m = 2 l°g [ V(* - «) + Vi* ~ /»)]• W 

In the second case, the sum of the factors being ft — a, we 
may put 



^r — a = (/? — a) sin 2 
/? — *=(/?_*) cos 2 



}■ 



whence 

dx = 2(/3 — a) sin cos #^#, 

and the quantity under the integral sign reduces to 2d6. There- 
fore 

(dx /x — a 

The same transformations may of course be employed when 
other factors occur in connection with these radicals 



70 METHODS OF INTEGRATION. [Art. 58. 

It can be shown that the values given in formulae (N) and 
(O) differ only by constants from the results derived by em- 
ploying the process given in Art. 56. 



Examples V. 

V(a — x)*x dx, {a — x)% ($x + 20) 

V(x + aj-x' 2 dx, - (a + x)% — — (a 4- x)% + —{a + x% 

J 7 5 3 

(x dx 2 3 
— , - x 2 — x 4- 2 a/x — 2 log ( 1 4- ifo). 
1 + Vx' 3 r & v r / 

J#d&c 2 , x ,, \ 

-77 r, - ix — 2CL) \/{x + tf), 

. j , X _ ■, . 2 1/^ + 2l0g(l — V*). 



6-f (. + ,)»,* 3^_^=-^. 

J-« 7 7 4 Jo 28 

(<£*• 2 _ x /2# — # 

#4/(20.* — a % y a V a 



8 . f(,_*)i*V*, -^ + 4^__^ 

Jo V 9 7 5 



_ 16a* 
y/a ~ 3*5 



[ *** x i £ , 3^ , 3** 1 3log(g**- . 

J 2*2 — # 3 44 8 



§ V.J EXAMPLES. fl 

Jo 8 5 Jx 10 4c 

12. -7- « s\ , Rationalize the denominator. 

Jx — V(x 2 — a) ' 



x' + jx" - a') 
3" 2 



2 (x + a)* — 2 (x 4- b)% 



f ^ 

I3 ' J V(x + a) + V(x + t) 9 3 (« ~ &) 

[ dx t i/(> 4 + 1) — 1 

I4 ' W> + «)' ~4 log ^R 



f |/(* 4 + 1) <&: j/(^+i) I V(-v 4 + 1) -1 

5 ' J x ' 2 4 g 



4 b ^(* 4 + 1) + I s 

Li v / (- v4 + - 1 

4 10g ^r 4 + i) + T 



16. 



f(. v"+i)(.y"-i)* ^ 
x 



ij-^O! + (^rjO 1 _(*._,)* + tan- ■ V{x n _ l} ] . 



X 

i/(jf a — # a ) — a sec - r - . 



^2 METHODS OF INTEGRATION. [Ex. V. 

f x % dx 

ig ' ]•(** + «■)■ 

= ,, a — ^-dx. See formulas (Z) and (K). 



-xV(x 2 + 2 ) - - 2 log [x + tf(x % + a')] . 



f V(a* - x>) _ .a- Y(a 2 - x*) % 

o. j — *-j <rfr, log K - '- + V(a % - x*). 

Jdx 
x + V{*' + a 1 ) ' 



x 



22 



V(* 2 + 0') + -log [x + V(x* + a )] 5 

20" 2 ° L X /J 20 



. ji^l^l ^ ^ Art 51. 

f 0Jtr 

log [ ^(.x + ) + *J 



x 



Jx dx t 

4 , . &* Formula {K\ -log |V + V(i + * 4 )], 



§ Voj EXAMPLES. 71 

25. \/{ax* + b) dx, [a >o] Put V{ax 2 + b) = z —x*/a. 

log [xVa + V(ax* +6)] +± xV(ax* +b). 



2 ya 



, f ^ 

J (« + x) V(x* + £ 2 ) ' 



Vd + 


*») 


Ji- 




ff 

-cotoT = 

6 


--Y3- 


2 2 

x — 2a 



1 * + ^ 2 + £ 2 ) + a - Vja' + b 2 ) 

V(a*+P) g x + V(^ a + b 2 ) + a + *V 4- £ 2 ) 



9 o f 1 ^ 

Ji*V(i-*V 

dx 

(P + qx)V{x* + 1)' 

V(/+rt l0g ^ ^[ tan_1 * + tan_I f] ' 
f *& V(# 9 — 1) . 1 

° J* |/(JC — i) 2X 2 

( a x 2 dx , 37T a/2 
7- log tan ^- — . 



74 METHODS OF INTEGRATION. [Ex. V. 



„ f * f .^ , . ^'-o 

33 ' J*V(*'-I)' ^ +I) 3*° • 

r ^ i /a + a? 

34 ' J(a-^)V(« a -^)' aVa-x' 

(dx i _ x ^4/2 

6 r« .**<&: r r* **<** -I 

3 °' Jo V{a-x) L Jo V(**-**)J 

2 

37> )xV(2ax-3*y ~a^ 



na 



2a — x 



a x 



° J^V(^ — i) 2X % 

39. y(2<2Jr — x )*dx 9 — 
Jo 4 

40. V{2ax — x*)'X dx, 

a 3 n cos 2 6 (1 + sin 0) </0 = « 3 - — I I 

2 

41. 4/(20.* — ^ 2 )-jt 2 ^jf, 

« 4 f° ff cos 8 (1 + sine) 2 dQ = a* [~^ - -~] 



§ V.] EXAMPLES. 75 

f dx 

42 ' J V( 2 ^r + x") ' 

by Art. 56, log [x + a + V(2ax + jc 2 )] + C; 

by Art. 58, 2 log [4/.* + 4/(20 + jt)] + C 

44. f/i^^[=Jy (2 ^^)]. 



. , X — - 

sin - 4/(20.*; — Jt: ), 



Jdx „ x — 2 
—-, — ■ jt , by Art. 56, sin- x V C: 



£y -4r/. 58, 2 sin- 1 |/^-±-l + £". 



4 6 - ~77 *T> 2 sin" 1 !/ - = ?r, 

J V{ax-x 2 )' * aj n 



'o 



f x 1 dx . x— 1 (x + 3) 4/(3 + 2* — * 2 ) 
47. J v(3 + 2 ,_, T 3sm-— 1. 



48 



* dx 7t 



49 



_ 2 4/(2 — at — •a: 2 )' 2 

[ 2a dx 
■L v{S-axy log( 3 + aVa). 



?6 METHODS OF INTEGRATION. [Ex. V. 

50. Find the area included by the rectangular hyperbola 

y = 2ax + x*, 
and the double ordinate of the point for which x = 20. 

a 3 [6 V2 — log (3 + 2V2)]. 

51. Find the area included between the cissoid 

x (x* + y 1 ) = lay* 

and the coordinates of the point (a, a) ; also the whole area between 
the curve and its asymptote. 

( — 71 — 2 J(Z 2 , and STta 1 . 

52. Find the area of the loop of the strophoid 

x{x> +/) + a (x*-f) = o; 
also the area between the curve and its asymptote. 

^(i-j). ^d 20' (1 + J). 



T^r /^<? loop put y = — x 3 3- , .sw^ a; 2> negative between the limits 

— a and o. 

53. Show that the area of the segment of an ellipse between the 
minor axis and any double ordinate is ab sm~ l h xy. 



§ vi. J 



INTEGRATION BY PARTS. 



77 



VI. 

Integration by Parts. 
59. Let u and v be any two functions of x ; then since 

d (uv) = u dv + v du y 

uv = udv + v du y 



whence 



\udv — 



uv 



v du 



• • (i) 



By means of this formula, the integration of an expression 
of the form udv, in which dv is the differential of a known 
function v, may be made to depend upon the integration of 
the expression v du. For example, if 



we have 



u = cos -1 ^ 



du 



and 
dx 



dv = dx, 



V{i-x*Y 



xdx 



hence, by equation (i), 

fcos" I ^-^r = ^cos _I ^ + 

in which the new integral is directly integrable ; therefore 

Qos- x x*dx = ^rcos" 1 ^ — V(i — x 2 ). 
The employment of this formula is called integration by parts. 



78 



METHODS OF INTEGRATION. 



[Art. 6a 



Geometrical Illustration. 



60. The formula for integration by parts may be geomet- 
rically illustrated as follows. Assum- 
ing rectangular axes, let the curve be 
constructed in which the abscissa and 
ordinate of each point are correspond- 
ing values of v and u y and let this 
curve cut one of the axes in B. From 
any point P of this curve draw PR 
and PS t perpendicular to the axes. 
Now the area PBOR is a value of the 




Fig. 2. 



manner the area PBS is a value of 
and we have 



indefinite integral u dv, and in like 
\vdu\ 



Area PBOR = Rectangle PSOR - Area PBS; 



therefore 



\u dv — uv — \v 



du. 



Applications. 



61. In general there will be more than one possible method 
of selecting the factprs u and dv. The latter of course in- 
cludes the factor dx, but it will generally be advisable to in- 
clude in it any other factors which permit the direct integra- 
tion of dv. After selecting the factors, it will be found con- 
venient at once to write the product u-v, separating the factors 
by a period ; this will serve as a guide in forming the product 



§ VI.] INTEGRATION BY PARTS. 79 

v du, which is to be written under the integral sign. Thus, let 
the given integral be 



J* log* 



dx. 



Taking x 2 dx as the value of dv, since we can integrate this 
expression directly, we have 

x 2 log x dx — log X' — x z Ix 3 — 

= — x 3 log x Ix^dx 



X 3 

= -(3 log*- i). 



62. The form of the new integral may be such that a 
second application of the formula is required before a directly 
integrable form is produced. For example, let the given 
integral be 

x 2 cos x dx. 

In this case we take cos x dx = dv ; so that having x 2 — u, the 
new integral will contain a lower power of x : thus 

x 2 cos x dx — ^-sin x — 2 \x sin x dx. 

Making a second application of the formula, we have 

Ix* cos x dx — X* sin x — 2\ x(- cos^r) + \cosxdx 

= x*smx + 2x cos x — 2 sin x. 



80 METHODS OF INTEGRATION. [Art. 63. 

63. The method of integration by parts is sometimes 
employed with advantage, even when the new integral is no 
simpler than the given one ; for, in the process of successive 
applications of the formula, the original integral may be repro- 
duced, as in the following example: 

e mx sin {nx + a) dx 

— cos (nx + a) m t , . , 

— s mx • i '- H s mx cos (nx + a) dx 

n n ) 

= ^ L + _ e mx ^ i e nx sin (nx + a)dx, 

n n n n l J ■ 

in which the integral in the second member is identical with 
the given integral ; hence, transposing and dividing, 

e mx s j n ( nx + a) d x — 2 \j n sj n (n X + a ) — n cos (nx + a)]. 

64. In some cases it is necessary to employ some other 
mode of transformation, in connection with the method of 
parts. For example, given the integral 

fsec 3 0ar<9; 

taking dv = sec 2 6 dd, we have 

f sec 3 a?0 = sec 0-tan - f sec tan 2 </0 . . . (i) 



§ VI.] FORMULAE OF REDUCTION. 8l 

If now we apply the method of parts to the new integral, by 
putting 

sec tan 6 dd = dv> 



the original integral will indeed be reproduced in the second 
member; but it will disappear from the equation, the result 
being an identity. If, however, in equation (i), we transform 
the final integral by means of the equation tan 2 6 — sec 2 6 — I. 
we have 



[sec 3 6dd = sec 6 tan 6 - [sec 3 6 dd + 
Transposing, 



sec Odd. 



2 jsec 8 6 dd : 



sinfl 
cos 2 6 



hence, by formula (F), Art. 31, 

f ■,-« sin 6 1 , \~7t 6~] 

sec 3 Odd = ^ + -log tan -+- . 

J 2 cos 2 82* |_4 2 J 

Formula of Reduction. 

65. It frequently happens that the new integral introduced 
by applying the method of parts differs from the given integral 
only in the values of certain constants. If these constants are 
expressed algebraically, the formula expressing the first trans- 
formation is adapted to the successive transformations of the 
new integrals introduced, and is called a formula of reduction. 



82 METHODS OF INTEGRATION. [Art. 65. 

For example, applying the method of parts to the integral 

\x n £ a *dx, 

we have 

If ax n f 
x n £ ax dx — x n \x n - 1 s ax dx ) .... (1) 

in which the new integral is of the same form as the given 
one, the exponent of x being decreased by unity. Equation 
(1) is therefore a formula of reduction for this function. Sup- 
posing n to be a positive integer, we shall finally arrive at the 

(8 aJC 
s a * dx, whose value is — . Thus, by successive appli- 
a 

r ntion of equation (1) we have 

r 7 e ax r n . x n(n — i) • • • • i~! 
\x n E ax dx = — x n x n - x - ... + (— iY -± 1 I . 



Reduction of \sin m 6 dd and [cos"' 6 dd. 



66. To obtain a formula of reduction, it is sometimes neces- 
sary to make a further transformation of the equation obtained 
by the method of parts. Thus, for the integral 

lsin m 8d6, 

the method of parts gives 

f sin m 6dd= sin w - I (9(-cos6') + (m — i) jsin™- 2 0cos 2 Odd. 



§ VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 83 

Substituting in the latter integral 1 — sin' 2 6 for cos 2 6, 

[sin™ Odd = — sin"'- 1 6 cos 6 

+ {111 - 1) I sin 7 "- 2 Odd — (m - 1) [sin w #^0: 
transposing and dividing, we have 
f sin- 6d 9 =- sin "'-' ° COS % -g— L [sin— grfO, . . . (ij 

a formula of reduction in which the exponent of sin 6 is dimin- 
ished two units. By successive application of this formula, we 
have, for example : 



' . B _ __ sin 5 6 cos 6 5 
sin 6 6d0= g + |- 



sitfddd 



sm 5 6 cos 6 5 sin 3 cos 6 5 3 f . , . ,. 

7 ■ — i f- 7- - sin 2 aT0 

6 64 64J 



sin 5 (9 cos 6 5 sin 3 # cos 6 5-3 sin cos 6 5-3-1 . 
6 6-4 6-4-2 6-4-2 

67. By a process similar to that employed in deriving 
equation (1), or simply by putting = \n — 6' in that equa- 
tion, we find 

f , n ?n cos'" -1 6 sin 8 m — 1 f ,„ „ „ ,,, , x 

cos w 6 d6 = + cos w - 2 ddd, . . (2) 

J mm) 

a formula of reduction, when ;;z is positive. 



84 METHODS OF INTEGRATION. [Art. 6& 

68. It should be noticed that, when m is negative, equation 
(i) Art. 66 is not a formula of reduction, because the exponent 
in the new integral is in that case numerically greater than the 
exponent in the given integral. But, if we now regard the 
integral in the second member as the given one, the equation 
is readily converted into a formula of reduction. Thus, put- 
ting — n for the negative exponent m — 2, whence 

m = — n + 2, 

transposing and dividing, equation (i) becomes 

f dd cos 6 n — 2 f dd ( , 

Jsin"6>~~ ~ (n— i) sin 7 *" 1 6 + n — I Jsin*- 2 ^' ' * " w 

Again, putting (9 = \ n — 6' in this equation, we obtain 

f dd sin 6> n — 2 [ dd 



cos n d (n — i) cos ; 



»-2[ flf^ 

/z— I Jcos«- 2 (9 * * ' 4 ' 



Reduction of \sin m d cos n d dd. 

69. If we put dv = sin wz cos # ^, we have 
cos n ~ x dsm m + z d 



sm m dcos n ddd = 



m + i 



+ - *- { sin>"+ 2 d cos n ~ 2 d dd ; . . . (i) 

w+ i J 

but, if in the same integral we put dv = cos w d sin ddd, we 

have 

sin^-^cos^ 1 d 



- 



sin™ d cos" d dd 



n + i 



m L [ s in~- 2 0cos« +2 0^. ... (2; 

»+i J 



§ VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 85 

When m and n are both positive, equation (1) is not a 
formula of reduction, since in the new integral the exponent 
of sin is increased, while that of cos is diminished. We 
therefore substitute in this integral 

sin w + 2 = sin™ (1 — cos 2 0), 

so that the last term of the equation becomes 



n — 1 
m + 1 



sin w cos* ~ 2 0^0 



n — 
m 4- 



H' 



sin'" cos" dO. 



Hence, by this transformation, the original integral is repro- 
duced, and equation (1) becomes 



[ 



I + 



n — 1 
m + 1 



sin w 0cos w 0</0 = 



sin^tfeos"- 1 ^ 
*« + 1 



VI 



— - [sin™ 
+ 1 J 



cos n -*ddd. 



Dividim by I H = , we have 

/^ + 1 w + 1 



sin w cos* ^0 



sin^^cos*- 1 ^ 



m + n 



n — 1 f . 
■ si 



s'm m 8 cos"' 2 6 d8, . 



(3) 



a formula of reduction by which the exponent of cos is 
diminished two units. 



80 METHODS OF INTEGRATION. [Art. 69. 

By a similar process, from equation (2), or simply by put- 
ting 6 = J 7t — 6' in equation (3), and interchanging m and n y 
we obtain 

sin*"- x 0cos* +1 



sin'* 6 cos" 8dd = 



m + n 



yyt T P 

+ sin'* - 2 cos" Odd, . . . (4^ 

m -t- n J v 

a formula by which the exponent of sin 6 is diminished two 
units. 

70. When n is positive and m negative, equation (1) of 
the preceding article is itself a formula of reduction, for both 
exponents are in that case numerically diminished. Putting 
— m in place of m y the equation becomes 

fcos" 6 Jn cos"" 1 6 n—i (cos n ~ 2 6 jn , x 

~ n dd—— -. r— ^ 75 -. -dd. . . (t) 

}sm m . (m—i)sm m - T 6 in—i)sm m - 2 6 °' 

Similarly, when m is positive and n negative, equation (2) gives 

fshv«0 , n sin 7 *" 1 /? m—i [sm m ~ 2 d ,_ fr . 

n do = n - Q dd . . . . (6) 

Jcos*# (n— i)cos n -*d n—\ J cos"" 2 6 w 

71. When m and n are both negative, putting — m and — n 
in place of m and n, equation (3) Art. 69 becomes 

f dS 1 



sin™ 6 cos* 6 {m + n) sm m ~ x 6 cos" +I 6 



+ 



n + 1 { 
m + n Jsii 



m + n i sin'* cos* +2 ' 
in which the exponent of cos 6 is numerically increased. We 



§ VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 87 

may therefore regard the integral in the second member as the 
integral to be reduced. Thus, putting n in place of n + 2, we 
derive 

f dd 1 



J sin w 6 cos* 6 {n — 1 ) sin w - l 6 cos" - z 

m + n— 2 r dft? , . 

« — 1 Jsin^^cos*- 2 ^ ^'' 

Putting 6 = \n — d' y and interchanging m and n, we have 

f </fl I 

J sin™ cos* # ~ (m—i) sin" 2 - x cos" - z 

/« + n — 2 f d# 
+ ;«— 1 Jsin w - 2 6>cos w ^ 



72. The application of the formulae derived in the preced- 
ing articles to definite integrals will be given in the next sec- 
tion. When the value of the indefinite integral is required, it 
should first be ascertained whether the given integral belongs 
to one of the directly integrable cases mentioned in Arts. 27 
and 28. If it does not, the formulas of reduction must be 
used, and if m and n are integers, we shall finally arrive at a 
directly integrable form. 

As an illustration, let us take the integral 

J sin 2 6 cos 4 d dd. 

Employing formula (4) Art. 69, by which the exponent of sin 6 
is diminished, we have 

f • 2 n 4 n jn Sin 6 COS° I f . Q Jn 

sin 2 8 cos 4 8 dd = -? + -? cos 4 6 dd. 



88 



METHODS OF INTEGRATION. 



[Art. 72. 



The last integral can be reduced by means of formula (2) Art. 
67, which, when m = 4, gives 



cos 4 e dd 



cos 3 6 sin 6 



2- fcos 2 fl^; 



therefore 



sin 2 cos 4 ddd=- 



sin #cos ! 



cos 3 #sin# . sin d cos 8 



6 24 

73. Again, let the given integral be 

f cos 6 fl ^ 
J sin 3 " 



16 



76' 



By equation (5), Art. 70, we have 



T cos 6 6 dd 
J sin 3 6> 



cos J 



2 sin' 



5 [cos 4 



Odd 



2 J sin 6 



We cannot apply the same formula to the new integral, since 
the denominator m— 1 vanishes ; but putting n — 4 and m = — 1, 
in equation (3) Art. 69, we have 



"cos 4 Odd cos 3 # 



sin 6 



3 

cos 3 6 

3 

cos 3 6 



cos 2 6d6 



+ 



sin 6 

r ^ 



sin # 



sin 6>d# 



Hence 



+ log tan — + cos 0. 
3 B 2 



fcos 6 0d?0 cos 5 6 5 cos 8 5 . 1 ^ 5 _ 

— r-g-^- = - ■ — r-ya — - — ^ log tan - d - - cos 0. 

J sin 3 2 sin 2 6 2 ta 2 2 



8 vi.] 



EXAMPLES. 



89 



I. I sin- 1 ^^, 



2. sec -1 ^^, 



(tan- 1 
o 

U w log^ 



xdx» 



dx 



5. * sin dQ, 

It 

6. r cos mBdQ i 

7. U^tan- 1 ^^. 

8. lx 2 e*dx 

Jo 

9. larsec- 1 *^, 



Examples VI. 



•arsin -1 * + V(i 



-*>!-';- 



xsqc 1 x — log [x + V^ 2 — i)J 



n log 2 
4 2 



£[*«*- rh] 



7T . W7T 2 . „ W^ 

— sin 7. sin" — . 

im 2 m 4 



1 + x 2 x 

tan -1 x 

2 2 



tf*** — 2X£ X +2 6" 



i [x 2 sec- 1 x — V(x* — 1)]. 



. J 2 sin - + e dfy — 6 cos (- + 6 J + sin ( -+ 0) 2 = ^— ■ 



METHODS OF INTEGRATION. [Ex. VI, 



ii. Ixsec'xdx* 



12. 



13. 



14. 



15- 



16. 



17. 



18. 



19. 



x tan x + log cos x. 



x tan 2 x dx 



= \x (sec 2 .* — 1) dx L ^tanjc + log cosx x 2 



x 1 sin x dx, 



2x sin x 4- 2 cos ^ — x cos jc. 



# sin - * x dx, - x* sin - x .%■ 



7T 

o 2 L 



sin' ode 



7t 



r 2 tan - x ^ ^c, 



^tan- 1 ^ „r a log (1 + x*) 



1 2 + x 

x* sin- x x dx, - x* sin - x jc H 4/(1 — •**) 

3 9 . 



^-^cos^^r, 



£■* tan ^ cos ^c dx, 



£~ x (sin x — cos x)~ 

2 



7t 2 

I 

^2 " 



cos/3€ xtsm Psm(/3 + x). 



8-*sin 2 xdx\ = - I €-■■*■ (1 — cos 2x)dx , 



— (cos 2jc — 2 sin 2X — k) 
10 v D/ 



20. 



J 4 £ sin 6 </0, 

J o 



(sin 6 — cos 



7T 



§ VI.] EXAMPLES. 91 

f E x 

21. \£ x sin * cos x dx, — (sin 2x — 2 cos 2x). 
J 10 

r . 4 « _*. — sin 3 wG cos 026 , 36 3 sin »z6 cos a*6 

22. sm ;«5 «5, ■ — \- — « . 

J o \m 8 Sm 

23. Derive a formula of reduction for \(\ogx) n x m dx, and deduce 
from it the value of (log.*) 3 x 2 dx. 

JX m + x 71 f 

(log x) n x™ dx = (log x)» (log x)"- 1 x fn dx. 

f/i N3 2 7 /, xs-** /1 xa-* 8 2A- 3 log* 2^ 3 

J(log^) 3 .rV.r= (log*) , j-(log.r) a j + -*--_. 

24. .v cos 2 a: dfc, j- j? sin jc cos -r — i sin 3 x + ^x 2 . 

25 . f^«ec-^ ( fa, AJsec - ,J: -- V ^- l) - l08[jr+ f J '- l)] . 

Ji 36 6 

26. Derive a formula of reduction for \x n sin (^ + a) dx, and de« 
duce from it the value of \x* cosxdx. 

\x n sin (x + a) dx = — jc» sin x + a -\ — 

+ # Ijc*" 1 sin Lr + « -i — dx. 

I* 5 cos x dx = (-* 5 — 2a# 8 + 120.*) sin ^ 4- (5^ 4 — 6a* 2 -f- 120) cos % 



9 2 



METHODS OF INTEGRA TIO.Y. [Ex. VI. 



J 



27. cos sin Qdd, 



sm 9 cos sin 9 cos 9 . 1 



7T 

28. 4 cos 4 6 sin 4 6 dQ, 



+ ^7 re — sin cose], 
24 16 



IT 

32 Jo 512 



29. 



f4 4a 1 sin cos 3 , 3sin0cos0 ,3914 8 + 3^ 



77 



30. 1 cos 6 d% 



31 



3 2 



sin cos (8 cos 4 + 10 cos 3 -f 15) + 150 

f cos4 e j« 
J sm 3 ' 

f sin2 6 j, 
J cos 5 ' 



33- 



34> 



3 _ 9V3 -t- io ?f 

o " 96 



cos 3 _ 3 cos _ 3 log tan |-9 
2 sin 2 B 2 2 



sin 9 sm 



Llogtan[j+|]. 



4 cos 4 8 cos 2 9 8 



(sin 7 sin 5 sin 5 r . ■ ^ 

— i— </0, r — — 3 — + ^- [0 — sin cos 0] . 

cos 3 cos 3 cos 2 L 



F s£» *, _ S2£lf - s F cos*e * = 48 ~ 's* , 

jTrsm 2 sin0 Jtt ^Jtt 32 



35- 



i<TT 



cos 0) s 



36. 1, 



7T 

1 17 afe' _ 2 

2 J o cos 4 9'"~ 3' 



1 1 , . 

3 cos 9 cos 9 2 



§ VI.] EXAMPLES. 



93 



37- J r ^ ,^ 2 „„ , T^XTX^ o 7^T + 5 log tan- . 



3 cos 6 3 9 

o • 2 „ + i log tan - 
4 sin 6 cos 6 8 sin e 8 & 2 



38. Prove that when # is odd 



aft sec*- 1 © , sec*- 8 6 , , . 

H h 4- log tan B ; 



j sin & cos" e « — 1 n — 3 
and when « is even 



dQ sec* _I , sec w_3 6 . 6 

H H + log tan - . 



39 



sin 6 cos w n — 1 n — 3 2 

[ dB 15 rsec 3 e „ , e"i 

• h~T7 i7> ^tt n +- +sec + log tan - L 

Jsmecose 2 sine cose 2 [_ 3 2 J 



V(*-i) ' Putx = ^«- 

\x«(x> - 1) + flog |> + V(x* - 1)]. 

J/ a ,i, (W — 2X 2 ) X^id* — A' 2 ) 7tf 4 . * 

(* - * T «**, — V^ ' + %" sm- 1 - . 

7T 

f^r 1 f7 . . 37T + 8 
r-3— ; — an » —r cos d$ = - 5- . 

43. j (a 2 + x*) V{a* - ^) dx, ^- sm- 1 - + — ^ ^ - J 

44- — 3 , 77— i ax + log \X + V(^T 2 — d 2 )\ 

f x* dx x(x t — 1) tan -1 a 

45 ' J(^ 2 + i) 8 ' S(x* + i) 2+ 8 



94 



METHODS OF INTEGRATION. [Ex. VI. 



, f cos B — sin Q . r f, , . ^ . cose — sine .1 

4 6 - / • „ ■ ru ^ 9 — K 1 + sin 6 cos 6) , . n , — , de L 

H J(sinQ + cos6) L J ' (sine + cos 0) J 



47. Derive a formula for the reduction of 



sin e cos e 
sine 4- cose 



sec* x dx ; and refer- 



ring to Ex. 11, thence show that this is an integrable form when n is 
an even integer. Give the result when n — 4. 



i x sec w x dx 



xsec n - 2 x tan x sec n ~ 2 x 



— 1 



(n— i)(n— 2) 



n — 2 



x sec n ~ 2 x dx. 



x sec 4 x dx 



6 3 



+ - \x tan x + log cos x]. 



48. Derive a formula of reduction for x cos** x dx, and deduce 
from it the value of x cos 3 x dx. 



, ^cos" _I .rsm^ cos^^r n — ^ 

\x cos* x dx— 1 = 1 x cos*" 2 x dx. 

n n n 



-J< 



, a* sin.*, „ x cos^f, 9 . 

\x cos .* tf.r = (cos x + 2) -\ (cos x + 6). 

3 9 



49. Find the area between the curve 

y = sec -1 .* 



VI.] EXAMPLES. 



95 



the axis of x, and the ordinate corresponding to x = 2. 

2 — -log [2 + y 3 ] = 0.77744. 

50. Find the area between the axis of x, the curve 

y = tan - x x, 
and the ordinate corresponding to x = 1. ^— ±= 0.43882 



VII. 



77^ Integral and its Limits. 

74. Before proceeding to some formulae of integration 
involving special values of the limits, it will be convenient to 
resume the consideration of the integral as defined in the first 
section. 

In Art. 3 we supposed a variable magnitude of which the 
values depend upon some independent variable x (but of which 
the expression as a function of x is as yet unknown) to vary 
simultaneously wither, while x passes at a uniform rate over a 
certain range of values. And it is assumed that the rate of 
the function is then expressed in terms of x and its assumed 
rate ; or, what is the same thing, that the relative rate of the 
function as compared with that of x is known. Denoting this 
relative rate by/(V), a known function of x, let F(x) denote the 
function under consideration ; then by the notation introduced 

F[x)= \f(x)dx (1) 



g6 METHODS OF INTEGRATION. [Art. 74 

F{x) may now be defined as the function whose derivative 
is f(x), or whose differential is f(x)dx ; hence, from this point 
of view, integration is the search for the indefinite integral, or 
the inverse of the process of finding the derivative of a func- 
tion, equation (1) implying nothing more than that 

d\F(x)\=f(x)dx (2) 

But because this equation was found to be insufficient to fix 
the values of the function F, we introduced in Art. 4 the nota- 
tion of the subscript or lower limit ; so that, when we write 



F(x) = ( /(*) dx, 

J a 



(3) 



we fully define the value of F(x) by implying the additional 
condition that F(a) ^= o. The integral thus modified is some- 
times called a corrected integral, because the indefiniteness 
arising from the unknown constant of integration has been re- 
moved. It remains " indefinite " in the sense that it is a func- 
tion of a variable x to which no special value has been assigned. 
It is to be noticed that, in many applications, the constant is 
determined (and the integral thus " corrected ") by the condi- 
tion that some other value (not zero) of F{x) shall correspond 
to a given value of x. These given simultaneous values of x 
and F(x) are called the initial values. 

75. As explained in Art. 4, the value of x to which corre- 
sponds the required value of the magnitude is known as the 
final value of x, and is used as the upper limit of the integral. 
Denoting it by b, we thus assume that, as the independent 
variable x passes from a to b, the function F(x) passes by con- 
tinuous variation from its initial to its final value ; and, when 
this is the case, we may write 



^f{x)dx = F(b)-F{a), 



(4) 



§ VII.] THE DEFINITE INTEGRAL. 97 

in which the function F is defined, not necessarily by equation 

(3), which implies F(a) = o, but by the more general equation 

(i), or simply by equation (2). 

We may now define the definite integral in equation (4) as 

the increment received by a variable whose differential is f(x) dx 

while x passes from the value a to the value b ; or, as the mag- 

dx 
nitude generated at the rate fix)— while x, with the arbitrary 

dx . s , . 

rate -=-, varies from a to 0. 
dt 

The condition mentioned above, that F(x) shall vary con- 
tinuously, requires that, starting from some finite initial value 
F(a), it shall not become infinite or imaginary for any value of x 
between a and b. The function Fix) cannot become infinite or 
imaginary unless its derivative /(;r) becomes infinite or imagi- 
nary. Hence the condition will be fulfilled if fix) remains 
real and finite for all such values of x. It is, of course, also 
assumed that there is no ambiguity about the value of fix) 
corresponding to any of these values of x. Cases in which 
such ambiguity might arise will be considered later. 

76. Since the rate of x is arbitrary, we may regard dx in 
the expression for the integral as constant, and in that case it 
must be regarded as positive or negative according as b is 
greater or less than a ; in other words, dx must have the sign 
of b — a. When fix) does not change sign for any value 
between a and b, we can thus infer the algebraic sign of the 
definite integral. 

Thus, if b > a and f(x) is positive for all values between 
the limits, F(x) is an increasing function of x, and x is itself 
increasing ; therefore the definite integral denotes a positive 
increment. For example, 

( n sin x 
dx 
o X 

denotes a positive quantity. Moreover, since in this case/(;tr) 



9 8 METHODS OF INTEGRATION. [Art. ?6. 

is less than unity for all values between the limits, the integral 
is obviously less than the increment received by x, that is, less 
than 7t. 

It is evident from these considerations that an interchange 
of the limits changes the sign of the integral ; thus, 



^f(x)dx = -^f{x)dx,, 



which agrees also with equation (4). Again, we infer from the 
same equation that 

|VW dx = ^ f{x)dx + ^f(x)dx, 

if c is between the limits a and b\ and this is also true when c 
is outside of these limits, provided the condition mentioned in 
the preceding article holds for the entire range of values of x 
implied by the several integrals. 

77. Returning now to the corrected integral, equation (3), 
we see that it is the same thing as 

\'f(x)dx, 

i a 

in which x stands at once for the upper limit and for the in- 
dependent variable which, in the generation of the integral, is 
conceived to vary from the initial to the final value. When 
there is any danger of confusion between these two meanings 
of x, it is well to use some other letter for the independent 
variable ; thus 

\/{b) dz 

has the same meaning as the expression above ; and in general 
it must be remembered that an integral is a function of its 
limits, and not of the variable which appears under the integral 
sign unless the same letter serves also to represent one of the 
limits. 



§ VII.] THE GRAPH OF AN INTEGRAL, 99 

Graphic Representation of an. Integral. 

78. A geometrical illustration was given in Art. 3, in which 
f(x) was taken as the ordinate of a curve, and the integral was 

in consequence represented by an area. We shall now employ 
another illustration, in which the ordinate y represents the in- 
tegral itself, regarded as a function of its upper limit. In other 
words, we shall consider what is called the graph of the func- 
tion (or graphic representation, employing rectangular coordi- 
nates), and shall regard this curve as derived from the expression 
for the integral, and not from the result of any process of 
integration. Putting 

y = ( f{x)dx, (I) 

J a 

we have 

g =/(*) = tan 0, (2) 

where has the same meaning as in Diff. Calc, Art. 26. 

Thus for every value of x we know the inclination 0of the 
curve to the axis of x. The notation also implies that when 
x = a, y — o. Starting, then, from the position (a, o), if we 
imagine the point (x, y) always to move in the proper direc- 
tion, a direction which changes as x changes, it will trace out 
a definite curve, so that the function y has a definite value for 
each value of x. 

79. As an example, let us take the function 



y 



f Sin X J f X 



a case in which the indefinite integration cannot be performed. 
Starting from the origin, the describing point must move in the 
varying direction defined by 

tan = (2) 



100 



METHODS OF INTEGRATION. 



[Art. 79. 



When x = o, we have tan = 1, or = 45 ° ; the curve 
therefore starts from the origin at this inclination. But, since 

equation (2) shows that tan 
decreases as x increases, the 
curve as we proceed toward the 
right lies below the tangent 
at the origin, as in Fig. 3. 
The ordinate y continues, how- 

ever, to increase (compare Art. 

76) until, at the point A, x 
reaches the value n for which 



Fig. 3. 

= 0. As x passes through this value, changes sign ; there- 
fore y has reached a maximum value. From x = n to x = 2zr 
tan is negative, hence y decreases; in this interval sin x in 
equation (2) goes through numerically the same values as before, 
but the denominator x is now much larger than before, hence 
it is plain that y does not decrease to zero. In like manner it is 
obvious that it will increase again from x = 2it to x = $7t, and 
so on, reaching alternate maxima and minima, which contin- 
ually approach an asymptotic value of y corresponding to 
x = 00 . The form of the curve for positive values of x is there- 
fore that represented in Fig. 3. For negative values of x the 
curve has a similar branch in the third quadrant. It is evident 
that we can in no case have a finite value of the integral when 
x = 00 , unless, as in this case, the quantity under the integral 
sign approaches zero as a limit when x increases without limit. 

80. When the graph or curve of the indefinite integral is 
drawn, the definite integral between any limits c and d is rep- 
resented by the increment of y in passing from the abscissa c 
to the abscissa d, and the condition given in Art. 75 requires 
that the curve shall be continuous between the points corre- 
sponding to the limits ; in other words, that these points should 
belong to the same branch of the curve. 

In the illustration above, f{x) remains real and finite for all 



§ VII.] 



THE GRAPH OF AN INTEGRAL. 



101 



values of x ; accordingly in this case all values of the limits 
are possible. But if f{x) becomes infinite or imaginary for 
any value of x between the limits, it will usually be found that, 
although the indefinite integral y — F(x) may have a finite 
value at each limit, the corresponding points will belong to 
separate branches of the curve, and therefore equation (4) of 
Art. 75 cannot be used. 

For example, if f(x) = x"*, which is infinite for x = o, we 
have for the indefinite integral 

rdx _ 1 
} x 2 x 

and y is also infinite when x = o. 

The graph of this integral is an equilateral hyperbola, by 
means of which we can represent, for example, the definite in- 
tegral between the limits 
— 2 and — 1 as the differ- 
ence BR between two values 
of the ordinate. But we 
cannot interpret in this case 
an integral between whose 
limits the value x = o lies 
(that is, with one negative 
and one positive limit), 
because the corresponding 
points would lie on dis- 
connected branches of the 
curve. 

It will be noticed that, 
for the same reason, if we 




Fig. 4. 



use the notation of the corrected integral we cannot represent 
the entire curve by a single equation ; thus the two branches 
as drawn in Fig. 4 have the separate equations 
r dx C x dx 

y =\-^ and y = )Jf* 



102 



METHODS OF INTEGRATION. 



[Art. 80. 



the latter expression denoting an essentially negative quan- 
tity. 

81. It may happen, however, that the indefinite integral 
remains finite for a value of x which makes f(x) infinite. In 
such a case, if the value of fix) remains real while x passes 
through this value, we shall still have a continuous curve and 
a continuous variation of the ordinate between limits which 

include this value of x. For 
example, when/"(V) = x~%, which 
is infinite for x = o, and is real 
for both positive and negative 
values, we have 




y 



=1 



dx 



3^ 



T= x 

:* 2 



+ C 



Fig. 5. 
through zero, and we can write, for instance, 



The curve which is drawn in 
Fig. 5 touches the axis of y and 
is real on both sides of it, form- 
ing a cusp. Thus the ordinate 
varies continuously while x passes 



J-ur* 2 J_ x 



in accordance with equation (4), Art. 75, as illustrated in Fig. 
5 by the difference of ordinates BR. 



Multiple-valued Integrals. 

82. When the indefinite integral is a many-valued function, 
the condition that it shall vary continuously while x passes 
from the lower to the upper limit requires the selection of 
properly corresponding values at the two limits. In illustra- 



§ vii.] 



MUL TIPLE- VA L UED INTEGRALS. 



I03 



tion let us construct the graph of the fundamental integral 

r dx • -1 / x 

In this case x can on'y vary between the values — 1 and 
-f 1. Proceeding from the origin, the curve, Fig. 6, has the 
inclination 45 , which is gradually in- 
creased to 90 when x = 1, at the point 
A, for which the value of y is \n. As x 
passes from o to — 1 a similar branch 
is described in the third quadrant, com- 
pleting the curve drawn in full line, the 
ordinate of which is in fact the primary 
value of the indefinite integral sin -1 .a: 
(Diff. Calc, Art. 54). But the complete 
curve j=sin -1 ^, or^=sin y, has another 
branch continuous with this at the point 
(1, Jar), as represented by the dotted 
line. In order that the point moving 
in accordance with the integral expres- 
sion shall describe this branch, we must 
suppose that x, after reaching the value 
unity at the point A, begins to decrease, 
and that, as dx thus changes sign, the radical 4/(1 — x*), which 
then becomes zero, also changes sign, so that the quantity 
under the integral sign remains positive, and y goes on in- 
creasing. Since the radical cannot change sign, while varying 
continuously, except when it passes through the value zero (at 
which time its two values are equal), it is only when x= ± I 
that x can return to its previous values without causing y also 
to return to the corresponding previous values; that is, with- 
out making the generating point return upon the path already 
described. 

We may thus, by the alternate increase and decrease of x 
between its extreme values, describe an infinite number of 




Fig. 6. 



104 METHODS OF INTEGRATION. [Art. 82. 

branches. In this generation of the integral the value of the 
radical undergoes periodic change of sign but is never am- 
biguous ; for in drawing the curve we assumed that in equa- 
tion (1) the radical had its positive value when x = o and 
y = o. This assumption determines the value of the radical 
for every other value of y\ it is, in fact, always equal to cosjy. 

83. In the direct application of limits to the integral (1), it 
is of course sufficient to limit the indefinite integral to its 
primary value, the radical being positive for all values of x be- 
tween the limits ; but, when the integral is the result of trans- 
formation, care may become necessary in the selection of the 
values at the limits. 

Consider, for example, the integral 

f^ 2 dz 

J _ x !/( 2 - zy 

which can be evaluated directly by formula (/') in the form 

sin -1 — =z 2—. But suppose we make the transformation 
1/2 J -x 4 

dx 

z 2 = 1 — Xj whence dz — - ,- and y(2 — /) = 4/(1 -f x). 

2.Z 

It is first to be noticed that, because the value of z is neg- 
ative at the lower limit, we should in the value of dz put 

dx 

z — — |/(i — x)\ thus dz = — -. — v and the result of trans- 

formation is 



f* 2 dz if 



dx 



V{i - x 2 )' 



The relation z* — 1 — x shows that as z increases from — I 
to o, x increases from o to 1, and the corrected integral in the 
second member (which is sin -1 x) increases from o to \n. 
Then as z further increases from o to \/2, x decreases from 1 
to — I ; but the integral continues to increase from \n through 



§VIL] 



MUL TIPLE- VA L UED INTEGRA LS. 



105 



values which are not " primary," reaching the final value \n at 
the upper limit, so that we have for the value of the second 
member of the equation ^(f n — o) = %n as before. 

84. To illustrate another mode in which an integral may 
acquire multiple-values, let us consider the graph of the fun- 
damental integral 

f d x 



= tan" 1 x. 



Proceeding from the origin at the inclination 45 , the curve 
approaches, as x increases without limit, an asymptote parallel 
to the axis of x at the distance 
\n. The similar branch de- 
scribed for negative values of 
x completes the full line in 
Fig. 7, representing the pri- 
mary values of tan -1 x, which 
are employed in any direct 
application of limits to the 
integral. But the complete 
curve y = tan -1 x consists of 
an unlimited number of branches which are repetitions of this 
curve at successive vertical intervals each equal to n, so that 
each asymptote is approached by two branches. 

Now, when the integral arises from transformation, it may 
happen that x passes through infinity, changing sign, and then 
the ordinate will pass without discontinuity of value from one 
branch to the next. For example, given the integral 




dd 



J cos 2 6 + 9 sin 2 6 
if we put tan 6 = x> this becomes 



5" 



see' dd 
4-9 tan a 6> ; 



dx 1 
= - tan -1 $x 

1+9* 3 - 



= - [tan' 1 (- 
3 



|/3) — tan- x oj. 



106 METHODS OF INTEGRATION. [Art. 84* 

As 6 passes from o to \n, x increases from o to -f co , and 

tan -1 2>x increases from o to \n. But as 6 further increases 

from \n to £ n, x changes sign and passes from — 00 to 

— \ 4/3. During this interval tan -1 $x still further increases 

from \n to f n, which is therefore in this case to be taken as 

the value of tan" 1 (— 1/3), while we take zero as the value of 

tan" T o. Hence 

55 
f6 du 2rt 



cos 2 6 + 9 sin 2 6 9 



Formula of Reduction for Definite Integrals. 

85. The limits of a definite integral are very often such as 
to simplify materially the formula of reduction appropriate to 
it. For example, to reduce 

j x n e~ x dx. 

J o 

we have by the method of parts 

\x n e-*dx = — x" e~ x + n \s- x x n ~ 1 dXo 



Now, supposing n positive, the quantity x n s~ x vanishes when 
x — o, and also when x = 00 [See Diff. Calc, Art. 107 ; Abridged 
Ed., Art. 91]. Hence, applying the limits o and 00, 



I x n s~ x dx = n\ x n -*s~ x dx. 



i>- 



§ VI L] FORMULAE OF REDUCTION. IO7 

By successive application of this formula we have, when n is 
an integer, 

x n a -x dx = n(n — i) 2 • I. 

86. From equation (1) Art. 66, supposing w > 1, we have 

IT IT 

f 2 sin- ddd = ^Z_I f 2 sin* -2 Odd. 
Jo m Jo 

If m is an integer, we shall, by successive application of this 
formula, finally arrive at f dd - - or \ sin 6 dd = 1, according 

Jo 2 Jo 

as m is even or odd. Hence 

if ^ is even, sin- 6 dd = v 7-^ — 7-^ , . . (/>) 

J m(m — 2) 22' v ' 

and if m is odd, f* sin- 6 dd = ^ ~ *)("* ~ 3) ' ' ' ' 2 . . . (P'\ 
J Q m(m — 2) I , 

87. From equations (3) and (4) Art. 69, we derive 

77 7T 

2 sin w 6 cos w (9 ^(9 = n "~ * | sin- cos n ~ 2 ddd, 

.0 m + n } 

J~~ ^2 T f2 
2 sin- cos* ^0 = sin™- 2 d cos" d dd. 
m + n i 



108 METHODS OF INTEGRATION. [Art. 87, 

By successive applications of the first of these formulae, we 
produce a series of factors in the numerator decreasing by in- 
tervals of two units from n — 1, and ending with 1 or 2, as in 
formulas (P) and (P f ). By the second formula we then pro- 
duce a like series of factors in the numerator beginning with 
m — I. The successive factors in the denominator produced 
in the process form a single series beginning with m -\- n and 
decreasing by intervals of 2, the final factor being greater by 
2 than the sum of the exponents in the final integral. Now 
this final integral will take one of the four forms 

IT IT IT IT 

[ 2 dd y [ 2 sin 6 d6, f 2 cos 6 dd, or ['sin d cos d dO, 

according as m and n are even or odd. Thus the final factors 
in the denominator are, in the four cases respectively, 

2, 3, 3, 4; 

while the values of the final integrals are respectively 

i#, 1, 1, h 

The result will therefore be the same if we carry the series 
of factors in the denominator in all cases to 1 or 2, and ignore 
the final integral except in the first case (m and n both even), 
when its value is \n. Therefore we may write, when m and n 
are positive integers, 

fan- & cos" 8 d6 = (^-i)(^- 3) •■•(«- 0^-3)-^ . {Q) 
J (m -h n){in + n — 2) v 

provided that each series of factors is carried to 2 or I, and a is 
taken equal to unity, except when m and n are both even, in which 
case (x = \-rt. 

This formula, of course, includes formulae (P) and (P r ). 



§ VII.] CHANGE OF INDEPENDENT VARIABLE. IO9 



Change of Independent Variable in a Definite Integral. 

88. A change of independent variable is frequently made 
for the purpose of simplifying the limits, rather than of chang- 
ing the form of the integral. For example, suppose the limits 
in formula (Q) to be multiples of \n by any consecutive integers 
k and k + 1. Putting 6 = \kn + 0, we have, according as k 
is even or odd, 

It IT 

2 sin™ 6 co$ H Odd = ± V s'm m <p cos" <p d<p, 



or 



1 (*+I)- »- 

2 sin w 6 cos M Odd — ± V cos'" sin M </0. 

But by formula (Q) the integrals in the second members 
have the same value ; hence the numerical value of the integral 
of formula (Q) is the same for every quadrant. The sign to be 
used is obviously that of sin™ d cos n 6 in the quadrant in ques- 
tion. We can thus readily determine the value of the integral 
when the limits are any multiples of \n. When m and n are both 
even, the sign is always -f- , and we have to multiply the result 
of formula (Q) by the number of quadrants. In other cases, we 
have two positive and two negative results while 6 describes 
one revolution. 

89. When the limits of an integral of the form considered 
above are o and \n, we may by introducing the double angle 
obtain the limits o and \n. Since, if = 26, 

sin d cos 6=% sin 0, sin 9 6=%(i—cos 0), cos 2 0=i(i 4- cos 0), 

the integral will, whenever m and n are either both even or both 
odd, be thus converted into one or more integrals of the same 



IIO METHODS OF INTEGRATION. [Art. 89. 

form, with integral values of the exponents, and having limits 
of the desired form. 

For example, by this transformation we obtain 

n it 

j 4 sin 6 6 cos 4 6d6 = -Lf 2 sin 4 <p (1 - cos 0) d<t> 



64 L 4" 2 2 5-ViJ 64L16 5 



64 L 4- 2 2 5 * 3 • 1 J 64L16 5 

90. A transformation which does not change, or which 
interchanges, the limits may sometimes reproduce the given 
integral together with known integrals, and thus lead to its 
evaluation. When the limits a and b of any integral are finite, 
they will be reversed by the substitution 

x = a -f- b — z. 
Thus 

[ ° fix) dx=- f/0 + b — z) dz - f f(a + b — z) dz , 

or, since it is indifferent whether we write z or x for the in- 
dependent variable in a definite integral, 

[ f(x) dx = [ /(a + £ - *) dx« 
For example, 
Pfl sin 4 dO = f V _ 0) sin 4 (ar - 6) d6 = [" (n - 0) sin 4 6 dd : 

hence 

2V d sin* 6 dd = Trf'sin 4 ^, 

and, by formula (P), 

3. 1 n $71* 



1: 



sin 4 BdO^Tt 



4.22 16 ' 



§ VII.] THE INTEGRAL AS THE LIMIT OF A SUM. 



II 



91. When the limits are o and oo , they will be reversed if 
we assume for the new variable the reciprocal or a multiple of 
the reciprocal of x. For example, if in 



-! og A dx 



f Jog 



we put x = — , whence dx — — - — ^-, we find 

y y 



f x 2 log a — log y j . C x dy 
5 , s J - dy = 2loga\ , ^ , - a; 



hence 



U = log tf -=-^ 5 = 



n log a 

2a 



con: 



The Integral regarded as the Limit of a Stim. 

92. We have seen in Art. 3 that, if the curve y =f(x) be 

istructed, the integral f(x) dx (in which we suppose f(x) 

to remain finite and continuous as 
x varies from a to b) is rep- 
resented by the area included by 
the curve, the axis of x and the 
ordinates corresponding to x = a 
and x = b. 

Let CD in Fig. 8 be the curve, 
and let the base of this area, AB y 
whose length is b — a, be divided 
into n equal parts. Denote the 
length of each part by Ax, so that 

b = a -f n Ax, 




Fig. 8. 



and erect an ordinate as in the figure at each point of division. 



112 METHODS OF INTEGRATION. [Art. 92. 

The whole area is thus divided into n parts, as represented by 
the equation 

[b P a + A* P a-\- 2.AX pb 

ydx = I ydx -t- ydx +.'...+ ydx. (i) 

J a J a J a + Ax J !> - Ax 

Now if we denote the values of j/ corresponding to a ■+■ Ax s 
a + 2 J^r, . . . £ by y l , y 9 , . . . y n , the rectangles y^x, y^x, . . . 
y n Ax, which are constructed in the figure, are approximations 
to the several terms of the second member of equation (i). 
Hence the sum of these rectangles, 

2yAx=y^x + y 9 dx -f . . . + y n dx,. ... (2) 

is an approximate expression for the integral in equation 
(1). When, as in Fig. 8, y increases continuously while x passes 
from a to b, the sum of the rectangles exceeds the curvilinear 
area, that is, 2 ydx exceeds the integral. 

If we take for the altitude of each rectangle the initial 
instead of the final value of y in the interval, we shall obtain in 
like manner an approximate expression, 

2'yAx = yjx + y x Ax + . . . + y n ^ Ax, . (3) 

which will, in this case, represent an area less than the curvi- 
linear one, so that 2'ydx is less than the integral. Thus the 
integral is intermediate in value to the two expressions in 
equations (2) and (3), of which the difference is 

2 ydx — ^2'yAx = ( y n — y Q )dx. . . . (4) 

Therefore the difference between the integral and either of 
the approximate expressions is less than {y n — y Q ) Ax. 

Now when the number of parts n is indefinitely increased, 



§ VII.] THE INTEGRAL AS THE LIMIT OF A SUM. IIj 

so that Ax is decreased without limit, the limit of (y n — y ) Ax 
is zero; it follows that \ ydx is the limit of 2 yAx for any 

range of values of x for which y is an increasing function of x. 
It can be shown in like manner that the same thing is true 
while y is a decreasing function, and therefore in general 

C 'j x = b 

f{x) dx — the limit of 2f(x) Ax . . . (5) 

J a x = a 

when Ax decreases without limit. 

93. The typical term/(;tr) Ax is called the element of the sum 
in equation (5). It will be noticed that the sum will have the 
same limit whether the x in f [x) Ax be taken as in equation 
(2) or as in equation (3), or in any other manner, provided it be 
some value within the interval to which Ax corresponds. The 
expression f(x)dx is, in like manner, called the element of the 
integral in equation (5). 

In many applications of the Integral Calculus, the expres- 
sion to be integrated is obtained by regarding it as an element. 
In so doing we are really dealing with the element of the sum ; 
but as we intend to pass to the limit we may, in accordance 
with the remark made above, ignore the distinction between 
any values of x between x and x + Ax. By equation (5), we 
pass to the limit by simply writing d in place of A, and the 
integral sign in place of that of summation,* so that in practice 
it is customary to form the element of the integral at once by 
writing d instead of A. 

* The term integral, and the use of the long s, the initial of the word sum, 
as the sign of integration, have their origin in this connection between the 
processes of integration and summation. 



114 METHODS OF INTEGRATION. [Art. 94. 

Differentiation of an Integral. 
94. The integral f(x)dx is by definition a function of x, 

J a 

whose derivative, with reference to x, is/(^). Thus, putting 

This gives the derivative of an integral with reference to its 
upper limit. By reversing the limits we have, in like manner, 

dU 

when the lower limit is regarded as variable. 
95. Now writing the integral in the form 



U 



u dx , o, . . . . , c (i) 



if u is a function of some other quantity, a, independent of x 
and a, U is also a function of a, and therefore admits of a de- 
rivative with reference to a. From (1) we have 

dU 

dx 
whence 

d dU _ du 
da dx da 

By the principle of differentiation with respect to independent 
variables [See Diff. Calc, Art. 401 ; Abridged Ed., Art. 200], 



§ VII.] DIFFERENTIATION OF AN INTEGRAL. I 1 5 





d dU _ d dU 




dx da da dx 


Therefore 






d dU _ du > 




dx da da ' 


and by integration 






dU {du , 
da J da 



(2) 



Now, in equation (i), U is a function of x and « which, when 
x = <?, is equal to zero, independently of the value of a. In 
other words, it is a constant with reference to a, when x — a\ 

therefore — — = o when x = a. If, then, we use a as a lower 
da 

limit in equation (2), we shall have C ' = o. Therefore 

-1.** » 



arc/ 



Substituting for x any value b independent of a, we have 



d_ 
da 



udx 



-—udx, ...... (4) 



which expresses that ## integral may be differentiated with 
reference to a quantity of which the limits are independent, by 
differentiating the expression under the integral sign. 

96. By means of this theorem, we may derive from an inte- 
gral whose value is known, the values of certain other inte- 
grals. Thus, from the first fundamental integral, 



x " dx = JTT-i' ' • • ' ' u » 



II 6 METHODS OF INTEGRATION. [Art. 96. 



we derive, by differentiating with reference to n, 

f . , in + i)^ +I log^ -x n + x 

x n log x dx = '- — ; ^ — - , 

J s (» + i) 2 

the result being the same as that which is obtained by the 
method of parts. 

97. The principal application of this method, however, is 
to definite integrals, when the limits are such as materially to 
simplify the value of the original integral. Thus, equation (1) 
of the preceding article gives 

f 1 I 

x n dx — , 

Jo »+l 

whence, by successive differentiation, 

I 



x n log x dx 



(n + if 



x*(io g xydx= ( ^ + 2 I)3 > 



X* (logx) r dx = (— i) r 



(n + i)> 



Integration under the Integral Sign, 

98. Let u be a function of x and a, and let a and a be con- 
stants ; then the integral 

U=\ ff udx\da, ........ r:» 



§ VII.] INTEGRATION UNDER THE INTEGRAL SIGN. II 7 

is a function of x and a, which vanishes when a — a , inde- 
pendently of the value of x, and when x — a, independently of 
the value of a. From (i) 

dU f , , d dU 

— - = \ u dx. whence -; — — =u\ 

da j a ax da 

n. ■ t ddU u dU f v , r 

therefore -— —- = u, whence -j- = \u da + C. 

da dx dx J 

Nowj- must vanish when a = a o , since this supposition makes 

£/ independent of x\ therefore, if we use a o for a lower limit 
in the last equation, we must have C = o\ therefore 



dU f i 

— — = u da. 

dx )a Q 
and since U vanishes when x = a, 

u =lil: da '\ dx - w 

Comparing the values of Urn equations (i) and (2), we have 

u dx da = \ u da dx. 
It is evident that we may also write 

Ia. 1 [b fb [Otx 

u dx da — udadx, . . . . (*) 

ocja )a)a o W 

provided that the limits are all constants. 



Il8 METHODS OF INTEGRATION. [Art. 99. 

99. By means of this formula, a new integral may be de- 
rived from the value of a given integral, provided we can inte- 
grate, with reference to the other variable, both the expres- 
sions under the integral sign and also the value of the inte- 
gral. Thus, from 

fl T 
x n dx = -, 
n+i' 

by multiplying by dn, and integrating between the limits * 
and s, we derive 

whence 

r 1 x s - x r , . s + I 

—z dx — log . 

J o log* *r+ 1 

100. When the derivative of a proposed integral with refer- 
ence to a is a known integral, we can sometimes derive its 
value by integrating the latter with reference to a. Thus, let 



u = — dx . (I) 

Jo X 



In this case 

du r , e-«n°° 1 

_ = -€-**<£* = — = --; 

da J a J a 

hence, integrating, 

u = — log a + (7= log-, (2) 

in which dThas been determined by the condition derived from 
equation (1) that, when a = p, u = o. 



§ VII.] ADDITIONAL FORMULAE OF INTEGRATION. II9 



Additional Formulce of Integration. 

101. The formulae recapitulated below are useful in evalu- 
ating other integrals. (A) and (A') are demonstrated in 
Art. 17; (B) and (C) in Art. 29; (D) and (£) in Art. 30; 
(F) in Art. 31 ; (G) and (G') in Art. 35 ; (H) and (/) in Art. 50; 
(/) in Art. 51 ; (AT) in Art. 52; (L) in Art. 53 ; (M) in Art. 55 ; 
(N) and (O) in Art. 58; (P) and (/") in Art. 86; and (Q) in 
Art. 87. 



( 



x — d)\x — b)~ a — b ^ x — b' 
ix I _ x — a f dx 



[ax I , x — a [ dx 1 <rz -f- ^r 

t ^ = — l°g — ; — ' ~9 9 = — log • • 

J x l — « 2 2# & x + « \ar — x* 2a ^ a — x 



[sin 2 #^ = \{B - sin cos 6) 



cos 2 6d6 = i(6 + sintfcostf). . 
40 



sin cos 6 



= log tan # 



^ 1 * 1 a 1 I - cos 

-t — 7, = log tan ifl = log : — - — 

sin 6 5 2 s sin 



(A') 
(C) 



tW 



f (W 

) cos 6 s 



-+-"] = log T -^ 1 ^ = log (sec 0+ tan 0). (F) 



-. 



+ bcosO ^(a 2 - £ 2 ) 



tan -1 



cos 6 
/a — b 



ta 



n i0~|. . . . (G) 



120 ADDITIOXAL FORMULAE OF INTEGRATION. [Art. lOI. 
f d6 _ I V(b + g) -u y(j - a) tan j- g 

\ .rV(c?-x>) =; l0S x ' < 7 > 

](«» + *)* = *•(*»» + *) •••(•/) 

( V(*» ± a 2 ) <& = * V V±<*) ± * l g [x + v(;r i ± a3)] . . (/ r ) 
f V (a 2 - ■*V.r = ysin-f +**(<*-*> (J/) 

J v[(x - a){x - m = 2 '°g [ ^- ») + <n* - m • • • w 

It 1t 

P sin- grfg = f ? cos" 0<fl? = < w ~ 'X*» - 3) • • • • i . ; (/J) 

Jo J m(m — 2) 2 2 ' 



§ VII.] ADDITIONAL FORMULAS OF INTEGRATION. 121 

— ■ — - — — — ■ ■»» 

p sin- dO = f cos- Odd = (^D^ - 3) "'\ . . {pf) 
Jo Jo m(ni-2) i 

(2 . „ „ „ rn (m — l)(m — i) • • X (fl — l)(fl— i) ' • - , ~* 

sin w 6cos n 6dd = ^-, A w D/ v , A ^ a, . (Q) 

J c (in -t- ?i) (tn + 71 — 2) v ' 



in which or = 1. unless m and « are both even, when a = — 

2 



Examples VII. 

t. : , [a > b, and n an integer] ~~m u\ 



( 2W,r± 2 di9 2n7t±_ \n 

2 + cos 9* 4/3 



\ Y sin 6 e^e, 

$. sin 5 Q d% 
5. ff cos 7 9d&, 



55 
32 

16 
IS' 

16 

3$ 



6 Psm 4 cos 6 6 afo, Ria 



122 METHODS OF INTEGRATION. [Ex. VII. 



7. sin 3 6 cos 4 6 dB, 

Z 1 

8. I sin™ 6 cos™ e dB, — I 

Jo ' 2™J C 



1 

10 



11 



f 1 x 2n+I dx 
■ Jo i/(i-^)' 



12 



r (**-<?)* dx 



r x° dx 
Jo (af + #*)* 



_4_ 
35 



sin™ 6 aft . 



f 1 x 2n dx 1-3-5 ' ' " (20 — 1) 7T 

Jo |/(i — x l ) y 2-4-6 • • • • 2n 2 



2«4'6- • • - 2fl 

4/(1-*') ' 3.5.7. . . (2* + 1) 



20 s ' 
63* 



^ 5 ^ _8_ 

(a 2 ** 2 )*' *5* 



p x 4 dx 
15. Prove that 

(a fa 

x n -*{a — x)™-* dx = I x>*-* (a—x) n ~> 
o o 



71 

32tf ? 



dx. 



and derive a formula of reduction for this integral, supposing n > o 
and m > 1. 

Stf ^ — j ta 
x n ~ 1 { a —. x }m- x fa — j x n { a ^ x yi - s ^ 



§ VII.] EXAMPLES. 



23 



16. Deduce from the result of Ex. 15 the value of the integral 
when m is an integer. 



* , i-2-v • • (m — 1) 



Jo 0(0+1) • • -{n+m—i) 

17. [" (0 + *) 5 (0 - a-) 1 ^v. 5"^ £*. 16. 

J - a 

IT 

18. Tsin 7 S (cos &) 2 </S. /W sin 2 5 — a, a?id see Ex. 16 



45°45 



tf 2 + x 1 40 



5. 7. 11. 19 

19. Show by a change of independent variable that 
r a- 3 /£v _ r <7 2 <fo 

Jo (a 2 + x>y -J («- + *») a ' 

and therefore -7-, an — — 

Jo (a + xy 2 . 

C°x\ogXjdx 

' Jo V + a 2 ) 2 ' 

[" tan- 1 .!-. afr 
• J a 2 + * + 1' 

I 



20 



21 



loga 



6V3' 



, jc a dx n 

22. I tan^ 



23. Derive a series of integrals by successive differentiation of the 

-X 

definite integral | s- ax dx. 

r , i-2- • • * 

jf 7r £-** dx = — 

Jo <** +I 



124 METHODS OF INTEGRATION. [Ex. VII. 



24. Derive from the result of Art. 6$ the definite integral. 

(e- mx sin nx dx — -^— — - 5 and £" mx cos nx dx — — 

m + n ' Jo /« 

and thence derive by differentiation the integrals 



xs w *sin nxdx — 7— j— - — ^ , and ^£" w * cos nx dx — 7—5 ^ 

(m -f /z a )" Jo 



m — n 

(m'+nj 



25. From the results of Ex. 24 derive 

x*e-»**smnxdx= v f — — -' ; 

Jo (W -f // ) 



1: 



(2 2\ 

»2 — -in ) 

26, From the fundamental formula (k 1 ) derive 






and thence derive a series of formulas by differentiation with refer- 
ence to a. 



f 00 dx _ n 1-3* •■(2« — 3) I 

o (a + fix 2 )* - ^M'i-2 ...(«- 1)' a— *" 



■27. Derive a series of integrals by differentiating with reference to 
o che integral used in Ex. 26, 



(°° x 2n ~ 2 dx n 1-3-5 •• (2/2 — 3) * 

(a + /?* a )« ~~ 2^i 1-2.3... («- 1) ' /J""* 



§ VI I J EXAMPLES. 125 

28. From the integral employed in examples 26 and 27, derive 

the value of 7 — - — tj-^t . 

J (a + A* 2 ) 4 

Differentiate twice with reference to /?, tf/zd' once with reference torot. 

1 

f°° .vVjc I-3-I 7T 

J o (a + /lr 2 ) 4 " ^3 ' i6ai/St ' 

29. Derive an integral by differentiation, from the result of Ex. II., 67 

dx n (2a 4- b) 



Jo (*" 



30. Derive an integral by integrating 



+ lr) (a 2 + «■)" 4 a J b (a + by 
dx 7C 



tan- l *- — tan- z - \ — = - log - , 
Jo L x x_\ x 2 * g 



31. Derive a definite integral by integrating 

r n 

€~ mx sm nx dx — — — t — = 
J m + n 

with reference to n. 

(cos ax — cos bx)dx = — log — 

Jo * ' 2 ° /« + tf 

32. Derive a definite integral from the integral employed in Ex. 31 
by integration with reference to m. 

J-riyw |" e .„_ ,. te -j ^ = [- tan _^ -tan-?]. 



126 METHODS OF INTEGRATION. . [Ex. VI I. 

$$. Derive an integral by integrating with respect to m 

r m 

£- mx cos nx dx = —7, 2 . 

Jo m + n 

e -ax _ e -bx I tf + n* 



r e~ ax — B~ bx , i , 
cos nx dx = — log 

J o X 2 



34. Derive an integral by integrating with respect to n the integral 
used in the preceding example. 



■ ( sin ax — sin &n ^r = tan - x — \ f . 

jo x m + ab 

35. Show by means of the result of Ex. 32 that, n being positive, 

J 00 • 



sin nx _ it 

dx = — 

j? 2 



36. Derive an integral by integration from the result of Ex. II., 67, 

f- [tan-* - tan- il-*L_- i l oc ^^ + ^ 

37. Evaluate log -3 -jjdx by the method of Art, 100. 

Jo X -\- 

rt (a — if)- 

38. Evaluate log 1 + -, log^dfo it a (log a — x 1. 



§ VIII.] PLANE AREAS. 12J 



CHAPTER III. 

Geometrical Applications— Double and Triple 
Integrals. 



VIII. 

Plane Areas. 

f02. THE first step in making an application of the Inte- 
gral Calculus is to express the required magnitude in the form 
of an integral. In the geometrical applications, the magni- 
tude is regarded as generated while some selected independ- 
ent variable undergoes a given change of value. The inde- 
pendent variable is usually a straight line or an angle, varying 
between known limits ; the required magnitude is either a 
line regarded as generated by the motion of a point, an area 
generated by the motion of a line, or a solid generated by the 
motion of an area. A plane area may be generated by the 
motion of a straight line, generally of variable length, the 
method selected depending upon the mode in which the 
boundaries of the area are defined. 

The Area generated by a Variable Line having a Fixed 

Direction. 

103. The differential of the area generated by the ordinate 
of a curve, whose equation is given in rectangular coordinates, 
has been derived in Art. 3. The same method may be em- 
ployed in the case of any area ge»^rated by a straight line 
whose direction is invariable. 



128 



GEOMETRICAL APPLICATIONS. 



[Art. 103. 



Fig. 9. 



Let AB be the generating line, and let R be its intersection 
with a fixed line CD, to which it is always 
perpendicular. Suppose R to move uni- 
formly along CD, and let RS be the space 
described by R in the interval of time, dt. 
Then the value of the differential of the 
area,' at the instant when the generating line 
passes the position AB, is the area which 
would be generated in the time dt, if the 
rate of the area were constant. This rate 
would evidently become constant if the generating line were 
made constant in length ; and therefore the differential is the 
rectangle, represented in the figure, whose base and altitude 
are AB and RS ; that is, it is the product of the generating line, 
and the differential of its motion in a direction perpendicular to 
its length. 

104. In the algebraic expression of this principle, the inde- 
pendent variable is the distance of R from some fixed origin 
upon CD, and the length of AB is to be expressed in terms 
of this independent variable. 

When the curve or curves defining the length of AB are 
given in rectangular coordinates, CD is generally one of the 
axes; thus, if the generating line is the ordinate of a curve, 
the differential is y dx, as shown in Art. 3. It is often, how- 
ever, convenient to regard the area as generated by some 
other line. 

For example, given the curve known as the witch, whose 
equation is 

y*x — 2ay* + A^x = o (1) 



This curve passes through the origin, is symmetrical to the 
axis of x, and has the line x = 2a for an asymptote, since 
x — 2a makes y — ± co . 

Let the area between the curve and its asymptote be re- 



§ VIII.] AREAS GENERATED BY VARIABLE LINES. 



129 



quired. We may regard this area as generated by the line 
PQ parallel to the axis of x, y being taken 
as the independent variable. Now 



PQ — 2a — x, 
hence the required area is 



(2a — x) dy . . . . (2) 



From the equation (1) of the curve, we 
have 

2a f 
X ~ /+ 4 * 2 ; 



whence 2a — x 



%a l 




Fig. 10. 



/ + 4 tf 2 ' 

and equation (2) becomes 

A=U Z [ 2 dy =4^ 2 tan-^1 = 4*0*, 



Oblique Coordinates. 

805. When the coordinate axes are oblique, if a denotes 
the angle between them, and the ordinate is the generating 
line, the differential of its motion in a direction perpendicular 
to its length is evidently sin a-dx ; therefore, the expression 
for the area is 



A = sin a \y dx. 



130 



GEOMETRICAL APPLICATIONS. 



[Art. 105. 



As an illustration let the area between a parabola and a chord 
passing through the focus be required. It is shown in treatises 
on conic sections, that the value of a focal chord is 




AB — 4^ cosec 2 « 



(1) 



where a is the inclination of the chord 
to the axis of the curve, and a is the 
distance from the focus to the vertex. 
It is also shown that the equation of 
the curve referred to the diameter 
which bisects the chord, and the tan- 



FlG. II. 

gent at its extremity which is parallel to the chord is 



f — 4a cosec 2 ol-x . 



(2) 



The required area may be generated by the double ordi- 
nate in this equation; and since from (1) the final value of 
y is ± 2a cosec 2 a, equation (2) gives for the final value of x 



Hence we have 



OR — a cosec 2 a. 



fa cosec 2 a 
y dx, 
o 



or by equation (2) 

A = 4\/a 



a coseCa 



tjx dx = 



Sa 2 cosec 3 a 



Employment of an Auxiliary Variable. 

(06. We have hitherto assumed that, in the expression 

A = ^ydx, 



§ VI 1 1 .] EMPLO YMENT OF AN A UXILIAR Y VARIABLE. 1 3 I 

x is taken as the independent variable, so that dx may be 
assumed constant ; and it is usual to take the limits in such a 
manner that dx is positive. The resulting value of A will 
then have the sign of y, and will change sign if y changes 
sign. 

It is frequently desirable, however, as in the illustration 
given below, to express both y and dx in terms of some other 
variable. When this is done, it is to be noticed that it is not 
necessary that dx should retain the same sign throughout the 
entire integral. The limits may often be so taken that the ex- 
tremityof the generating ordinate must pass completely around 
a closed curve, and in that case it is easily seen that the com- 
plete integral, which represents the algebraic sum of the areas 
generated positively and negatively, will be the whole area of 
the closed curve. 

107. As an illustration, let the whole area of the closed 



curve 



& + ©' 



ii 

represented in Fig. 12, be required. If in this equation we put 

we shall have 

whence x = a sin 3 ip, and y — b cos 3 ip. • . (1) 

Therefore \ydx= lab cos 4 ip sin 2 ip dtp. 



32 



GEOMETRICAL APPLICATIONS. 



[Art. 107. 



Now if in this integral we use the limits o and 2?r, the point 

determined by equation (1) de- 
scribes the whole curve in the 
direction A BCD A. Hence we 
have for the whole area 




A 



= 3«* , 



cos 4 ip sin 2 1/> dip, 



6-4-2 



and by formula (0 

3-I-I * nah 

8 * 



The areas in this case are all generated with the positive 
sign, since when y is negative dx is also negative. Had the 
generating point moved about the curve in the opposite direc- 
tion, the result would have been negative. 



Area generated by a Rotating Line or Radius Vector. 



108. The radius vector of a curve given in polar coordinates is 
a variable line rotating about a fixed extremity. The angular 

rate is denoted by -=- and may be re- 
garded as constant, but then the rate at 
which area is generated by the radius 
vector OP, Fig. 13, will not be constant, 
since the length OP is not constant. 
The differential of this area is the 
area which would be generated in the 
time dt, if the rate of the area were con- 




Fig. 13. 



stant ; that is to say, if the radius vector were of constant 



§ VIII.] AREAS GENERATED BY ROTATING LINES. 133 

length. It is therefore the circular sector OPR of which the 
radius is r and the angle at the centre is dd. 



Since 



arc PR-r dd, 



sector OPR = -r> dd\ 
2 



therefore the expression for the generated area is 

A=±[r*d0 .... 



109. As an illustration, let us 
find the area of the right-hand loop 
of the lemniscata 



(0 




r 1 — a* cos 2d. 



Fig. 14. 



The limits to be employed are those values of 6 which 

make r = o ; that is and — . 

4 4 

Hence the area of the loop is 

it n 

A=-(* cos 2ddd = l sin 2d\\ = 

2J-L 4 J-* 



2 ' 



110. When the radii vectores, r 2 and r x corresponding to the 
same value of 6 in two curves, have the same sign, the area 
generated by their difference is the difference of the polar areas 
generated by r x and r 2 . Hence the expression for this area is 



A=l\{ri-r?)dd. . . 



(2) 



134 



GEOMETRICAL APPLICATIONS. 



[Art. Ill 



III. Let us apply this formula to find the whole area 
between the cissoid 



r x — 2a (sec B — cos 0), 

Fig. 15, and its asymptote BP 2 , whose 
polar equation is 

r 2 = 2a sec B. 

One half of the required area is generated 
by the line P\P^ while 6 varies from o to 
1 




Fig. 15. 



n. Hence by the formula 

jr 

A = 2tf 2 J o 2 (2-cos 2 6>) dd = $-7ta\ 
Therefore the whole area required is 372^ 



Transformation of the Polar Formulas. 

112. In the case of curves given in rectangular coordinates^ 
it is sometimes convenient to regard an area as generated by a 
radius vector, and to use the transformations deduced below 
in place of the polar formulas. 



Put 



y — MX', 



(1) 



now taking the origin as pole and the initial line as the axis 
of x, we have 

x = r cos 8 y y = r sin 6 ; . , . (2) 



therefore 
and 



~y 



yi — - — tan B, 
x 



dm = sec 2 9 dB. 



(3) 



§ VI 1 1.] TRANSFORMATION OF THE POLAR FORMULAS, 135 
From equations (2) and (3), 

x* dm = r 2 dd ; 

therefore equation (1) of Art. 108 gives 

A — - 1 x 1 dm, (4) 

In like manner, equation (2) Art. no becomes 

A=\\(xi-x*)dm (5) 

113. As an illustration, let us take the folium 

x 3 + f— $axy = o (1) 

Putting y = mx, we have 

x 9 ( 1 + m*) — ^amx 2 =0 (2) 

This equation gives three roots or values of x, of which two 
are always equal zero, and the third is 

X =J!^L; ....... ( 4 ) 

I + m % v ^ 7 



whence y= — „ (5) 

y 1 + m s KJJ 

These are therefore the coordinates of the point P'm Fig. 16. 
Since the values of x and y vanish when m = o, and when 
m = 00 , the curve has a loop in the first quadrant. To find 



136 



GEOMETRICAL APPLICATIONS. 



[Art. 113. 



the area of this loop we therefore have, by equation (4) of the 
preceding article, 

A _9 a% C w& dm 3a 2 1 ~T _ 3^ 

2 J (i + m s ) 2 ~ "21 + m s J ~" 2 



114. The area included between this curve and its asymp- 
p tote may be found by means of equation 
(5), Art. 112. The equation of a straight 
line is of the form 



y = MX + by 



0\ 

c 




Fig. 16. 



and since this line is parallel to y — mXy 
the value of m for the asymptote must be 

that which makes x and y in equations (4) and (5) infinite ; 

that is, m = — I ; hence the equation of the asymptote is 



y + x — b, 



(6) 



in which b is to be determined. Since when m= — I, the 
point P of the curve approaches indefinitely near to the asymp- 
tote, equation (6) must be satisfied by P when m— — I. 
From (4) and (5) we derive 



y -f- x = ia 
whence, putting m = — 



m a + m 



^am 



m 



;/r 



1 + nf 

, and substituting in equation (6) 
— a = by 
he equation of the asymptote AB, Fig. 16, is 

y + x = —a. 



• (7) 



§ VIII.] TRANSFORMATION OF THE POLAR FORMULAS* 137 



Now, as m varies from — 00 too, the difference between the 
radii vectores of the asymptote and curve will generate the 
areas OB C and ODA, hence the sum of these areas is repre^ 
sented by 



A =- 



(xi — x?) dm, 



in which x 2 is taken from the equation of the asymptote (/; 
and x x from that of the curve. 

Putting y — tux, in (7), we have 



x % 



1 + m 



and the value of x x is given in equation (4). Hence 



rr. 



gm 2 



3\2 



.(1 + nif (1 + m 3 ) 



] 



dm 



-?t 



+ m 



2 4- m — iw 
1 + m* 



a 2 2 — m ~l° * 



2 1 — m + m 



]o If 



= ^, 



Adding the triangle OCD, whose area is ^a 2 , we have for thu 
whole area required f<2 2 . 



* This reduction is given to show that the integral is not infinite for the 
value m = — 1, which is between the limits. See Art. 75« 



138 GEOMETRICAL APPLICATIONS. [Ex. VIII. 

Examples VIII. 

1. Find the area included between the curve 



a 9 y = x s + ax 1 , 



and the axis of x. 

2. Find the whole area of the curve 

a y = x* (a* - x*). 

3. Find the area of a loop of the curve 



12 



*>(*'+/)=/(*'-/). J (71-2). 

4. Find the area between the axes and the curve 

,& + <» = ?(—*). *[J-*p]. 

5. Find the area between the curve 

xy + ay — ax = o, 
and one of its asymptotes. 2a*. 

6. Find the area between the parabola y 1 = ^ax and the straight 

Sa* 
line y = x. — . 

7. Find the area of the ellipse whose equation is 



ax* + 2bxy + cf = 1. S(J-?y 



§ VI II.] EXAMPLES. I39 

8. Find the area of the loop of the curve 

cf=(x-a){x-b)\ 

m which r > o and b > a. — ~- • 

15^ 



9. Find the area of the loop of the curve 

a y = x 4 (b + x). 



3 2 ^ 
10502 



10. Find the area included between the axes and the curve 



©'♦©*- 



ab_ 
20 



11. If n is an integer, prove that the area included between the 
axes and the curve 



(f) 



<v' - • 



. n (n — 1 ) • • • 1 , 

is A — — — ^ r-^ — - — ■ — c ab. 

2n \2n — 1) •••(«+ 1) 



12. If n is an odd integer, prove that the area included between 
the axes and the curve 



©'+©*- 



_ \n{n— 2) - - • i] 2 7tab 
2n (211 — 2) • • • 2 2 



HO GEOMETRICAL. APPLICATIONS, [Ex. VIIL 

13. In the case of the curtate cycloid 

x = aip — b sin ip, y = a — b cos tp, 

find the area between the axis of x and the arc below this axis. 



{2a 2 + b 2 ) cos- 1 *; - 3 a V(b* - a % ) 

14. If b = iart, show that the area of the loop of the curtate 
cycloid is 



»•[?-]■ 



15. Find the area of the segment of the hyperbola 

x = a sec tp, y — b tan tp i 

cut off by the double ordinate whose length is 2b. 

ab\ 4/2 — log tan ^- L 

16. Find the whole area of the curve 

r 2 = a 2 cos 2 4- b 2 sin 2 6. - (a 2 + b*). 

2 

17. Find the area of a loop of the curve 

2 'ii 72 ■ 1 . Q>b (a 2 — b 2 ) _ T a 

r* = a 1 cos 2 — b 2 sm 2 0. — + tan l T „ 

22 b 

18. Find the areas of the large and of each of the small loops ot 
the curve 

r = a cos cos 20 ; 



§ VIII. ] EXAMPLES. I4I 

and show that the sum of the loops may be expressed by a single 
integral. 

na 2 a 2 . na 2 a 2 

—r + - , and . 

16 6 32 12 

19. In the case of the spiral of Archimedes, 

r — aS, 

find the area generated by the radius vector of the first whorl and 
that generated by the difference between the radii vectores of the nth. 
and (n + i)th whorl. 

— — and %nan % . 
3 

20. Find the area of a loop of the curve 

r = a sin 36. 

a 1. Find the area of the cardioid 

r = 4a sin 2 ^Q. 6rta*. 

82. Find the area of the loop of the curve 

cos 20 a 2 (4 — n) 

r = a -. — — -, 

cos B 2 

23. In the case of the hyperbolic spiral, 

re — a, 

show that the area generated by the radius vector is proportional to 
the difference between its initial and its final value. 



12 



H 2 GEOMETRICAL APPLICATIONS. [Ex. VIII, 

24. Find the area of a loop of the curve 



r = a cos « S. 



4/2 



25. Find the area of a loop of the curve 



r * = £ sin 38 ^ £ 

cos' 9 * 2 



26. Find the area of a loop of the curve 



Notice that r is real and finite from 9 — — to 9 = — , #«*/ that I 

4 4 J sin ^ 

& negative in this interval. # 2 4/2 — log (1 + Y2) L 

27. Find the area of a loop of the curve 

(x* +/Y = a'xj, 

tip 
Transform to polar coordinates. — 

4 

28. In the case of the limacon 

r — 2a cos 9 + b, 

find the whole area of the curve when b > 2a and show that the same 
expression gives the sum of the loops when b < 2a. 

(2a* + P)7t. 



§ VI 1 1.] EXAMPLES. I43 

29. Find separately the areas of the large and small loops of the 
limacon when b < 2a. 

If a — cos -1 ( — — ) , 

large loop = (2a 2 + F) a + ^ V(^ - <* 2 ) 5 
small loop = (2a 2 + £ 2 ) (n - a) - ^ 4/(4^ - £ 2 ). 



30. Find the area of a loop of the curve 

r 2 = a 2 cos 72 tf + & sin « 9. 

31. Find the area of the loop of the curve 

2 cos 2 # — 1 



i/(a 4 + ^ 



r = a 



cos 



[5V3-{-> 



32. Show that the sectorial area between the axis of x, the equi- 
lateral hyperbola 

x> -/ = 1, 

and the radius vector making the angle Q at the centre is represented 
by the formula 

. 1 _ 1 + tan 

and hence show that 

£2A _|_ f -2A £2A £-2A 

jtr = , and y = . 



If A denotes the corresponding ared in the case of the circle 
we have 



x* +/= 1, 



x = cos 2 A, and j = sin 2 A. 



144 GEOMETRICAL APPLICATIONS. [Ex. VIIL 

/;/ accordance with the analogy thus presented, the values of x and y given 
above are called the hyperbolic cosine and the hyperbolic sine of 2 A. Thus 

= cosh (2A), = sinh {2A), 



33. Find the area of the loop of the curve 

x A — ^axf + 2ay 3 = o. 

34. Find the area of the oop of the curve 



36. Find the area of the loop of the curve 



f + ax 1 — axy = o. 



37. Find the area of a loop of the curve 

x* + y* — a*xy. 

38. Trace the curve 

• y 

x— 2a sin-, 



3 a 
35"*' 



297 ~\~ I 

^2» + i +j,2» + i — ( 2 ^ + 1) ax n y n . a* 



35. Find the area between the curve 

2» + I a 

and its asymptote. a 



60 



and find the area of one loop nc? 



IX.] 



VOLUMES OF GEOMETRIC SOLIDS. 



145 



IX. 



Volumes of Geometric Solids. 



115. A geometric solid whose volume is required is fre- 
quently defined in such a way that the area of the plane sec- 
tion parallel to a fixed plane may be expressed in terms of 
the perpendicular distance of the section from the fixed plane. 
When this is the case, the solid is to be regarded as generated 
by the motion of the plane section, and its differential, when 
thus considered, is readily expressed. 

116. For example, let us consider the solid whose surface is 
formed by the revolution of the curve APB> Fig. 17, about 
the axis OX. The plane section per- 
pendicular to the axis OX is a circle; 
and if APB be referred to rectangu- 
lar coordinates, the distance of the 
section from a parallel plane passing 
through the origin is x, while the 
radius of the circle is y. Supposing 
the centre of the section to move 
uniformly along the axis, the rate at 
which the volume is generated is not 
uniform, but its differential is the vol- 
ume which would be generated while the centre is describing 
the distance dx, if the rate were made constant. This differen- 
tial volume is therefore the cylinder whose altitude is dx, and 
the radius of whose base is y. Hence, if V denote the volume, 




dV = 7tj? dx. 

117. As an illustration, let it be required to find the volume 
of the paraboloid, whose height is k, and the radius of whose 
base is b. 



146 geometrical applications. [Art. 117. 

The revolving curve is in this case a parabola, whose equa- 
tion is of the form 

f = 4<zx ; 
and since y — b when x — h, 

b 2 — ^ah, whence 40 = - ; 

to 

the equation of the parabola is therefore 



?=h x - 



Hence the volume required is 

V= 7t y % dx — 71 



x dx =- . 



118. It can obviously be shown, by the method used in 
Art. 116, that whatever be the shape of the section parallel to 
a fixed plane, the differential of the volume is the product of the 
area of the generating section and the differential of its motion 
perpendicular to its plane. 

If the volume is completely enclosed by a surface whose 
equation is given in the rectangular coordinates x, y, z, and if 
we denote the areas of the sections perpendicular to the axes 
by A x , A y , and A zy we may employ either of the formulas 

V = \a x dx, V = \A y dy y V=\a 8 dz. 

The equation of the section perpendicular to the axis of x 
is determined by regarding x as constant in the equation of 
the surface, and its area A x is of course a function of x. 



§ IX.] VOLUMES OF GEOMETRIC SOLIDS. 1 47 

For example, the equation of the surface of an ellipsoid is 

x 2 j/ 2 £ 

u J — + _ — 1 

a 2 ^ P c 2 



The section perpendicular to the axis of x is the ellipse 

f- £ __ a 2 — x % 



whose semi-axes are- y(a 2 — x 2 ) and - V(a 2 — x*). 



Since the area of an ellipse is the product of n and its semi« 
axes, 

a* 

The limits for x are ±a, the values between which x must lie 
to make the ellipse possible. Hence 



IF \ 



(a 2 - x*) dx =^ ad - C 



119. The area A x can frequently be determined by the con- 
ditions of the problem without rinding the equation of the 
surface. For example, let it be required to find the volume of 
the solid generated by so moving an ellipse with constant 
major axis, that its center shall describe the major axis of a 
fixed ellipse, to whose plane it is perpendicular, while the ex- 
tremities of its minor axis describe the fixed ellipse. Let the 
equation of the fixed ellipse be 

x* f 

— 4- — = T 



148 



GEOMETRICAL APPLICATIONS. 



[Art. 119. 



and let c be the major semi-axis of the moving ellipse. The 
minor semi-axis of this ellipse is y. Since the area of an 
ellipse is equal to n multiplied by the product of its semi-axes, 
we have 



ncy — — )/(a 2 — x 2 ). 



7tbc [ a 
Therefore V= — V(a 2 — x*)dx; 

hence, see formula (M), 

ri*abc 



The Solid of Revolution regarded as Generated by a 
Cylindrical Surface, 

120. A solid of revolution may be generated in another 

manner, which is sometimes more 
convenient than the employment 
of a circular section, as in Art. 1 16. 
For example, let the cissoid POR, 
Fig. 18, whose equation is 



Ri 


1 


? 




--i 




/ 






A 






~c 


P 




/ 
/ 



V 



Fig. 18. 



pass from the value OA 

will evidently generate the solid of revolution 



jp (2a — x) — x?, 

revolve about its asymptote AB. 
The line PR, parallel to AB and 
terminated by the curve, describes 
a cylindrical surface. If we con- 
ceive the radius of this cylinder to 
: 2a to zero, the cylindrical surface 

Now every 



§ IX.] DOUBLE INTEGRATION. 149 

point of this cylindrical surface moves with a rate equal to 
that of the radius; therefore the differential of the solid is 
the product of the cylindrical surface, and the differential o ( 
the radius. The radius and altitude in this case are 

PC - 2a - x, and PR = 27, 

therefore V — ^n [ {2ax — x*)^x dx. 

Putting x — a = a sin 6, 

IT 

V= 47ra 3 j 2 ^(cos 2 6 + cos 2 6 sin 6) dS = 2ttV. 

2 

Examples IX. 

1. Find the volume of the spheroid produced by the revolution of 

the ellipse, 

x* y 

47tad 2 



about the axis of x. 

3 

2. Find the-volume of aright cone whose altitude is a, and the 
radius of whose base is b. Ttab 2 



3. Find the volume of the solid produced by the revolution about 
the axis of x of the area between this axis, the cissoid 

y (2a — x) = x* t 
and the ordinate of the point (a, a). %a s n (log 2 — -f), 



150 GEOMETRICAL APPLICATIONS. [Ex. IX. 

4. Find the volume generated by the revolution of the witch, 

y*x — 2 ay 2 + ^c?x = o, 
about its asymptote. 

See Art. 104. 47rV. 

5. The equilateral hyperbola 

2 8 2 

■* — f = tf* 

revolves about the axis of x : show that the volume cut off by a plane 
cutting the axis of x perpendicularly at a distance a from the vertex 
is equal to a sphere whose radius is a. 

6. An anchor ring is formed by the revolution of a circle whose 
radius is b about a straight line in its plane at a distance a from its 
centre : find its volume. 2n' i at? i . 

7. Express the volume of a segment of a sphere in terms of the 
altitude h and the radii a x and a a of the bases. 

8. Find the volume generated by the revolution of the cycloid, 

x = a (ip — sin tp), y = a (1 — cos tp), 

about its base. 57rV. 

9. The area included between the cycloid and tangents at the 
cusps and at the vertex revolves about the latter ; find the volume 
generated. 

10. Find the volume generated by the revolution of the part of the 
curve 

y = e* 9 

which is on the left of the origin, about the axis of x. 

it 

2 ' 



§ IX.] EXAMPLES. 151 

11. The axes of two equal right circular cylinders, whose common 
radius is a y intersect at the angle a ; find the volume common to the 
cylinders. 

The section Parallel to the axes is a rhombus. 16a 3 

3 sin a ' 

12. Find the volume generated by the revolution of one branch of 
the sinusoid, 

about the axis of x. n^atf 



13. Find the volume enclosed by the surface generated by the revo- 
lution of an arc of a parabola about a chord, whose length is ze, per- 
pendicular to the axis, and at a distance b from the vertex. 

IS-"' 

14. Find the volume generated by the revolution of the tractrix, 
whose differential equation is 

* = 4. y 

dx * Vicf-yY 

about the axis of x. 

Express ny 2 dx in terms of y. 27za* 



15. Find the volume generated by the curve 

xy 1 = 4a (2a — x) 
revolving about its asymptote. 47rV. 

16. Express the volume of a frustum of a cone in terms of its 

height h, and the radii a l and a % of its bases. 

7th , -> 

— (a: + a x a, + a 2 "). 



152 GEOMETRICAL APPLICATIONS. [Ex. IX. 

17. Find the volume of a barrel whose height is 2^, and diameter 
2b, the longitudinal section through the centre being a segment of an 
ellipse whose foci are in the ends of the barrel. 

2 3F+IT 

18. Find the volume generated by the superior and by the inferior 
branch of the conchoid each revolving about the directrix ; the 
equation, when the axis of y is the directrix, being 

x y = (a + xYtf - ■*'). 

no? ± ^ 
3 

19. The area included between a quadrant of the ellipse 

x — a cos 0, y = b sin 0, 

and the tangents at its extremities revolves about the tangent at the 
extremity of the minor axis ; find the volume generated. 

rrab^io — 3JT) 



20. An ellipse revolves about the tangent at the extremity of its 
major axis ; express the entire volume in the form of an integral, 
whose limits are o and 2 7T, and find its value. 2 7rV*?. 

'21. Find the volume generated by the revolution of a circular 
arc whose radius is a about its chord whose length is 2c. 

^ — 27T0 1 a/ (a — <r 2 ) sin -1 -. 

3 a 

22. A straight line of fixed length 2c moves with its extremities 
in two fixed perpendicular straight lines not in the same plane, and 
at a distance 2b. Prove that every point in the moving line de- 
scribes an ellipse in a plane parallel to both the fixed lines, and find 
the volume enclosed by the generated surface. 47c (<r 2 — tf)b 



§ X.] DOUBLE INTEGRALS. 153 

X. 

Double Integrals. 

321. Let us consider the expression 

<P(x,y)dy, (i) 

J y\ 

in which the limits y l and jj/ a may be any functions of x. In 
the integration, ^ is to be treated like any other quantity 
independent of y which may be involved in the expression 
<p{x, y) ; in other words, as a constant with respect to the 
variable y. Thus the indefinite integral will contain both x 
and y ; but since the definite integral is a function not of the 
independent variable but of the limits, the expression (i) is 
independent of y, but is generally a function of x. We may 
therefore denote it by f(x), and write it in place of f{x) in the 
expression of an integral in which x is the independent vari- 
able. Thus, putting 

*0(x,y)dy=f{x), (2) 

y* 

a' y 



f(x)dx=\ (P{x,y)dydx (3) 



In the last expression, which is called a double integral, x 
and y are two independent variables. It is to be noticed that 
the limits of the ^/-integration, which is to be performed first, 
may be functions of the other variable x, but the limits of the 
final integration must be constants, that is, independent both 
of x and of y. 

122. In accordance with Art. 92, the expression (1) is the 

limit of 2 *(ft(x, y) Ay, when Ay is diminished without limit, 

bjing assumed that (p{x, y) is finite and continuous while y 
passes from y 1 to y 2 . Further, assuming this to be the case 



54 



GEOMETRICAL APPLICATIONS. [Art. 122. 



for all values of x from a to b (so that/(^) in equation (3) is 

finite and continuous for all values concerned in the ^-integra- 

b 
tion), the double integral is the limit of 2 f(x) Ax, when Ax 

is diminished without limit. It readily follows that the double 
integral is the limit of 

,^2 



2 a 2 y l ^ X> y) Ay AX ' 

where both Ay and Ax are diminished without limit.* 

The typical term <p{x,.y) Ay Ax is called the element of the 
sum, and in like manner (p(x, y) dy dx is the element of the doable 
integral As mentioned in Art. 93, in forming the expres- 
sion for the element, no distinction need be drawn between any 
values of x and y, between x and x -\- Ax,y and y + Ay, re- 
spectively, because these distinctions disappear at the limit. 

Limits of the Double Integral. 

123. In discussing the limits of the double integral (3), 
Art. 121, it is convenient to consider 
first the simpler expression 









1 








E / 




I 




/ 






/ 






[ 


c 


D 









J. 


1 


1 


3 





J aj y. 



dx. 



(1) 



Performing the ^-integration, this re- 
duces to the simple integral 



Fig. 19. 



\{y, —yd 

J a 



dx. 



(2) 



* Denoting the difference between f(x) and 2 q <p{x, y)Ay, of which it is the 
limit, by e, e is a quantity which vanishes with Ay. Then the difference be- 
tween 2 f(x) Ax and 2^2^ <p(x, y) Ay Ax is HjAx, a quantity which vanishes 

with Ay if a and b are finite. Hence, in this case, the double integral which is 
the limit of the first of these sums is also the limit of the second. But the con- 
clusion does not follow when the limits are infinite ; in fact, the double integral 
is not then always independent of the mode in which the limits become infinite. 



§ X.] LIMITS OF THE DOUBLE INTEGRAL. I 55 

In Fig. 19, using rectangular coordinates, let OA = a, 
OB = b, and let CD, EF be the curves y = y lt y = y % \ then 
(2) is, by Art. 103, the expression for the area CDFE. There- 
fore the double integral (1) is represented by the area enclosed 
by the curves y = y lf y = y^ and the straight lines x = a, x = b. 

124- . In order that the double integral may represent the 
area enclosed by a single curve (like the dotted line of Fig. 
19) of which the equation is known, the limits y l and y Q must 
be two values of y corresponding to the same value of x, and 
the limits for x must be those values for which y l and y % are 
equal. Between the limiting values of x the values of y are 
real, and beyond them the values of y become imaginary. For 
example, suppose the curve to be the ellipse 

2x 2 — 2xy -f y 2 — 4X — 2y -f- 6 — o ; 
solving for^, 

y = x + 1 ± |/(-* 2 + 6x — 5) = x .+ 1 ± ^[(x - i)(5 - *)] ; 
whence 

y 9 -y 1 = 2 i /[(x- i)($-x)l 

This expression is real for all values of x between I and 5 ; 
hence the entire area is 

A = ('pdydx = 2V tf[(x — i)(5 - x)] dx* = 4^. 

It is evident that we might equally well have used the ex- 
pression 



dxdy=\ (x.-x^dy, 



* It is useful to notice that a definite integral of this form, by a familiar 
property of the circle, represents the area of the semicircle whose diameter n- 
the difference between the limits. 



I56 GEOMETRICAL APPLICATIONS. [Art. 1 24. 

in which x x and x^ are obtained as functions of y, by solving 
the equation of the curve for x, thus 

* = *0' + 2)±*V(-.y + 87-8); 

and the limiting values of jy are those obtained by putting the 
radical equal to zero, namely, / = 4 — 2 \/2, q = 4 -(- 2 4/2. 
Accordingly the same area is expressed by 

'4 + 2V2 



f4-t-2V2 
J 4 - 2^2 



whence A = \n as before. 

125. It appears therefore that no distinction is to be drawn 
between the expressions 



dy dx and \dxdy. 



We may in fact regard either one as representing any area 
whatever, the value becoming definite only when we assign a 

defined closed contour or boundary line; just as \dx may 

represent a line of any length measured along the axis of x, 
and is definite only when we assign the limits which determine 
its two extremities. Thus the contour bears the same relation 
to the double integral that the pair of limits does to the simple 
integral. 

When the boundary of a given area is made up of lines 
having different equations, it is not generally possible to ex- 
press the area by means of a single double integral. This was 
possible, it is true, in the case of the area enclosed by the full 
line in Fig. 19, because the equations of two of the bounding 
lines, x = a and x = b, contained only one variable, and the 
integration with respect to this was made the final one in ex- 
pression (1). But, if integration with respect to x had been 
performed first, it would have been necessary to break the area 



§X.] 



THE AREA OF INTEGRATION. 



157 



up into several parts, of which it is the algebraic sum. In 
the expressions for these parts, the limits for x would be taken 
from the equations of the different lines, and the final limits 
would be the ordinates, which it would be necessary to find 
(instead of the known abscissas), of the intersections C, D, E, 
and F. 

126. Returning now to the general double integral 



I *<i>(x,y)dydx, 



(•) 



and employing rectangular coordinates, let the area ASBR 
in the plane of xy, Fig. 20, be that 
which is defined by the limits of 
the given integral, in other words, 
the area represented by 



rv 3 



dydx. 



(2) 




In evaluating the double integral 
(1) with the given limits, the integra- 
tion of the element <p(x, y)dy dx is 
said to extend over the area ASBR 
represented by expression (2). 

Now let us construct the surface whose equation is 

z = <p(x, y) 



(3) 



The ^-coordinates of all points whose projections on the plane 
of xy are on the curve ASBR lie in a cylindrical surface, which 
in Fig. 20 is supposed to cut the surface (3) in the curve 
CQDP. It is assumed that (p{x,y), or z, remains finite and con- 
tinuous while the independent variables x and y vary in any 
way, provided the point (x, y) remains within the area ASBR. 
Tin's assumption is clearly identical with that made in Art. 122. 
A definite solid will then be enclosed between the surface (A 



158 GEOMEI^RICAL APPLICATIONS. [Art. 1 26. 

the plane of xy y and the cylindrical surface. We have now to 
show that this solid will represent the double integral (1). 

Let SRPQ be a section of this solid by a plane parallel to 
that of yz, so that in it x has a constant value. The ordinates 
of the points R and 5 are the values of y % and y^ correspond- 
ing to that particular value of x. The section SRPQ may be 
regarded as generated by the line z, while y varies from y l to 
y % ; hence, denoting its area by A x , as in Art. 118, 

A x — \ z dy. 
hi 

Therefore the volume is 

V= [ b A x dx = I'Tzdydx 
which is identical with expression (1). 

Change of the Order of Integration. 

127. It is obvious that the order of integration may be 
reversed as in Art. 124, provided that the integration extends 
over the same area. Considering the corresponding process of 
double summation, we may be said, in the first case, to sum up 
those of the elements z dy dx which have a common value of x 
into the sum A x dx, which then constitutes the element of the 
second summation ; while in the second case we first sum up 
those of the original elements which have a common value of 
y into the element A y dy, which is afterward summed for all 
values of y within the given limits. 

Since, in expression (1), Art. 126, dx is a constant with 
respect to the ^-integration, it may be removed to the left of 
the corresponding integral sign. In the resulting expression, 



[ dx\ *4>{x,y)dy, 



§x.] 



CHANGE OF ORDER IN INTEGRATION 



159 



it is to be understood that, because the ^/-integral which is a 
function of x follows the sign of ^-integration, it is "under" 
it, or subject to it. This notation has the advantage of mak- 
ing it perfectly clear to which variable each integral sign cor- 
responds.* 

128. The limits of a double integral are sometimes given 
in the form of a restriction upon the values which the inde- 
pendent variables x and y can simultaneously assume. Sup- 
pose, for example, that in the integration neither variable can 
become negative or exceed a, and 
that y cannot exceed x. If x and y 
are coordinates of a point in Fig. 21, 
the restriction is equivalent to saying 
that the point cannot be below the 
axis of x, to the right of the line x = a 
or above the line y = x. Therefore 
the area of integration is the triangle 
OAB. This may be expressed either 
by giving to y the limits zero and x, 
and then giving to x the limits zero 

and a ; or, reversing the order of integration, by giving to x the 
limits y and a, and then giving to y the limits zero and a. 
Thus the area of integration can be represented by either of 
the expressions 



Fig. 2i 



dydx> or dx dy. 

129. As an illustration, let us take the integral 
U = II sin' 1 y 4/(1 — x) \/{x — y)dy dx. 

J oJ o 

The ^-integration, which is here indicated as the first to be 

* We have in the preceding articles supposed the first written integral sign 
to correspond to the last written differential when the pair of differentials is 
written last ; but writers are not uniform in this respect. 



l6o GEOMETRICAL APPLICATIONS. [Art. 1 29. 

performed, namely, the integration of \/{x — y) sin - y dy is 
impracticable. Noticing, however, that the ^-integration in 
the given expression is of a known or integrable form, we change 
the order of integration, determining the new limits so as to 
represent the same area of integration. Since this area is that 
indicated in Fig. 21, when a = 1, we thus obtain 

U— sin _I j/^ \/[(i — x)(x — y)~]dx. 

Jo J y 

The value of the ^-integral is \n{i — yf ; hence 

n r 
[/ — -\ sin x y(i — yfdy. 

Finally, integrating by parts and then putting y = sin 6, 

24 Jo 4/(1-/) 2 4 J V ; 24L4 3 J 

Triple Integrals. 
130. An expression of the general form 

n\<t>(x,y,z) dzdydx (1) 

is called a triple integral. In the first integration the limits 
z x arid £ 2 are in general functions of x and y. In the next, the 
limits jj/, and y^ are in general functions of x\ and in the final 
integration the limits are constants. 

It is readily seen that the triple integral, like the double in- 
tegral considered in Art. 122, is the limit (where Ax, Ay and Az 
diminish without limit) of the result of a corresponding sum- 
mation. Accordingly <p (x,y, z) dz dy dx is called the element 
of the integral (1). 



§ X.] TRIPLE INTEGRALS. l6l 

Consider now the triple integral with the same limits, but 
having the simpler element dz dy dx ; the ^-integration can 
here be effected at once, and we have 

^^dzdydx=\ "\z 2 - z,) dy dx. . . . (2) 

Now supposing x, y and z to be rectangular coordinates in 
space, the last expression represents the difference between 
two solids defined as in Art. 126; that is, the triple integral in 
equation (2) represents the volume included between the surfaces 
z = z i% z = £ a , and the cylindrical surface zvhose section in the 
plane of xy is tJie contour determined by the limits for y and x. 

131. When the volume is completely enclosed by a surface 
defined by a single equation or relation between x,y and z, the 
case is analogous to that of the area discussed in Art. 124; z x 
and #„ will be two values of z corresponding to the same values 
of x and y, and the limits for x and y must be determined by 
the area within which the value of z^ — z k is real. This area 
is the section with the plane of xy of a cylinder which circum- 
scribes the given volume. 

In this case it, is plain that we may perform the integrations 
in any order, provided we properly determine the limits. 
Considering the corresponding process of summation, we may 
be said in equation (2) to have summed those of the ultimate 
elements which have common values of x and y into the linear 
eleme?it (z u — z x ) dy dx. If the integration for y is effected 
next, we combine such of these last elements as have a com- 
mon value of x into the areal element A x dx which is often 
called a lamina. Thus each of the formulae in Art. 118 is the 
result of performing two of the integrations in the general 
expression for a volume, namely, 

V= [\\dxdydz. 



l62 GEOMETRICAL APPLICATIONS. [Ex. X. 

In any case where the result of two integrations (that is, the 
area of a section) is known, we may of course take advantage 
of this and pass at once to the simple integral, as in Art. 118, 
where in finding the volume of the ellipsoid we regarded the 
expression for the area of an ellipse as known. 

Examples X. 

i. Find the volume cut from a right circular cylinder whose 

radius is a, by a plane passing through the centre of the base and 

making the angle a with the plane of the base. 2 , 

& & r -a 3 tan a. 

3 

2. Show that the integral of x dx dy over any area symmetrical 
to the axis of y vanishes. Interpret the result geometrically, and 
apply to find the integral of {c +mx 4- ny)dx dy over the ellipse 

«y + £V = a 2 b\ nabc. 

3. Show that the volume between the surface 

z n = a v + dy 

and any plane parallel to the plane of xy is equal to the circumscrib- 
ing cylinder divided by n + 1. 

4. Find the volume enclosed by the surface whose equation is 

x u y a z* Znabc 

-*+ii+74 = I - — — ■ 

a b c 5 

5. A moving straight line, which is always perpendicular to a 
fixed straight line through which it passes, passes also through the 
circumference of a circle whose radius is a, in a plane parallel to the 
fixed straight line and at a distance b from it; find the volume en- 
closed by the surface generated and the circle. ncfb 

2 

6. A cylinder cuts the plane of xy in an ellipse whose semi-axes 

are a and b, and the plane of xz in an ellipse whose semi-axes are a 

and c, the elements of the cylinder being parallel to the plane of yz ; 

find the volume of the portion bounded by the semi-ellipses and the 

rtirface. 2 

-aoc. 

3 



§ X.] EXAMPLES. 163 

7. Find the volume enclosed by the surface 

2 2 
y z _ x 

oca 



and the plane x = a. 

8. Find the volume enclosed by the surface 

1,2,2. 2 

x* + y 3 + z* = a*. 

Find A z as in Art. 107. 



abc 
2 



PdMf- 



47ta 
35 * 

9. Find the volume between the coordinate planes and the surface 

abc 
90' 

10. Find the volume cut from the paraboloid of revolution 

y + 2 a = /±ax 

by the right circular cylinder 

x" 2 + y = 20.*, 

whose axis intersects the axis of the paraboloid perpendicularly at 
the focus, and whose surface passes through the vertex. 

2na H . 

3 

11. Find the volume cut from a sphere whose radius is a by a 

right circular cylinder w T hose radius is b, and whose axis passes through 

the centre of the sphere. 4 7zr F 3 / 2 72 \ 3 " 

— a — [a — ys 
3 L 

12. Prove that the volume generated by the revolution about the 
axis of y of the area between the equilateral hyperbola 

222 
x — y = a 

and the double ordinate 2y ] is equal to the sphere of radius y v [Ex. 
11 shows that the circle has the same property.] 



164 GEOMETRICAL APPLICATIONS. [Art. 1 32. 

XL 

The Polar Element of Area. 

132. When polar coordinates are used, if concentric circles 
be drawn, corresponding to values of the radius vector r hav- 
ing the common difference Ar, 
and then straight lines through 
the pole corresponding to values 
of the angular coordinate 6 having 
the uniform difference Ad, any 
given portion of the plane may be 
divided, as in Fig. 22, into small 
areas. The value of any one of 
these is readily seen to be Ar . rA6, FlG - 22 - 

where r has a value midway between the greatest and least 
values of r in the area in question. It follows that the result 

22rArA8 

of a double summation of this element between proper limits 
will give an approximate expression for the given area. Hence 
the double integral 




ilrdrdd, (1) 



which (compare Art. 122) is the limit of this expression when 
Ar and Ad are both diminished without limit, is the exact ex- 
pression for the given area. 

133. The differential expression rdrdd, or polar element of 
area, is the product of the differentials dr and rdO, which cor- 
respond to the mutually rectangular dimensions Ar and rAOoi 
the element of summation. These factors are the differentials 
of the motions of the point {r, 6), respectively, when r alone 



§ XL] THE POLAR ELEMENT OF AREA. l6$ 

varies and when alone varies. It is obvious that the ele- 
ment of area for any system of coordinates can in like manner 
be shown to be the product of the corresponding differential 
motions, provided only that these motions are at right angles 
to one another* The element is in such a case said to be 
ultimately a rectangle. 

The formula used in section VIII is in fact the result of 
performing the integration for r in formula (i) above. Thus 
when the pole is outside of the given area, as in Fig. 22, we 
have 



\^rdrdd = ^(r*-r*)d6, 



the formula of Art. no. The limits for 6 are now the values 
which make r % = r l ; that is, in the case of a continuous 
curve, the values for which the radius vector is tangent to the 
curve. So also when, as in the example of Art. 109, the pole 
is on the curve (so that r x = O for all values of 6), the limits for 
correspond to tangents to the curve. But, when the pole is 
within the curve, we assume the lower limit zero to avoid nega- 
tive values of r, and then make 6 vary through a complete revo- 
lution, that is, from o to 27T. 

Cylindrical Coordinates. 

134. In determining the volume of a solid, it is sometimes 
convenient to use the polar coordinates r and 6 in place of x 
andjj/, while retaining the third coordinate z perpendicular to 
the plane of rd. The system r } 6, z is sometimes called that 
of cylindrical coordinates, because the locus of r = a constant 

* In other words, whenever the loci of constant values of one coordinate cut 
orthogonally the like loci for the other coordinate ; as in the case of the coordi- 
nates latitude and longitude on a spherical surface, or on a map made on any 
system in which the representatives of the parallels and meridians cut at right 
angles. 



1 66 GEOMETRICAL APPLICATIONS. [Art. 1 34. 

is the surface of a right circular cylinder. The element of 
volume in this system is obviously the product of dz and the 
polar element of area, that is, 

rdr dd dz, 

which has the ultimate form of a rectangular parallelopiped. 

When this element is used in finding the volume of a solid 
bounded in part by a cylindrical surface of which the elements 
are parallel to the axis of z, like that represented in Fig. 20,. 
Art. 126, the integration for z is naturally performed first, its 
limits being the values of z in terms of r and 6 given by the 
equations of the surfaces which cut the cylindrical surface. 

135. For example, let us find the volume cut from a sphere 
of radius a by a right circular cylinder having a radius of the 
sphere for one of its diameters. Taking the centre of the 
sphere as origin, the diameter of the cylinder as initial line, 
and the axis of z parallel to the axis of the cylinder, we have, 
for the equation of the sphere, in these coordinates 

£ + r % = a 2 , (1) 

and, for that of the cylinder, 

r = a cos 6 (2} 

The limits for z are now taken from equation (1), those of r 
are zero and that given by equation (2), and those of 6 are ± \n 
which make r in equation (2) vanish. Hence we have for the 
required volume 

.- f a cos 9 t V(« 2 - r») 

V=~- \ dd\ rdr\ dz 

J _!T Jo J_ V (^._ r2) 

2 

(fa cos 9 (• - j— — itfcos0 

' d6\ (a*-r i ) , 'rdr = 2y dey-W-**?^ ■ 



§ XL] CYLINDRICAL COORDINATES. 1 67 

From the symmetry of the solid it is apparent that we may 
take the limits for 6 to be o and \it, and double the result.* 
Thus 



3 n 

V=^i 2 [i- sin 3 V)d6 = 
3 Jo 



27ra 3 Sa 3 

~3~ '~q' 




Solids of Revolutio7i with Polar Cooj'dinates. 

136. The volume of the solid produced by the revolution 
about the initial line of a curve given in polar coordinates may 
be expressed by a double integral of which the element is 
derived from the polar element of 
area. In this revolution, every 
point P of the polar element of 
summation, r Ar Ad, constructed 
in Fig. 23, describes a circle whose FlG - 2 3- 

radius is PR = r sin 6, and moves always in a direction per- 
pendicidar to the plane of the element. Therefore the volume 
described by the element is equal to its area multiplied by 
2nr sin 0, where r and belong to some point within the ele- 
ment. Since the distinction between all such points disap- 
pears at the limit, we have for the element of volume 

27ir 2 sin 6 dr d0\ 

and the integral of this expression over the given area is the 
required volume. 



In this example, the result, which would otherwise have been written in 



2a° f 2 



-a 3 
the form V = — [ (1 — sin 3 G)</&, would not have been correct, because the 

3 J-* 

2 

(a 2 — r 2 ) 3 which occurs in the integration stands for the positive value of the 
radical, whereas in the fourth quadrant a sin 0, which we have put for it, stands 
for the negative value. 



1 68 GEOMETRICAL APPLICATIONS. [Ait. 1 37. 

137. For example, the curve in Fig. 23 being the lemniscata 
r* = a 2 cos 20, 

the volume generated by the right-hand loop, or rather by the 
half-loop in the first quadrant, is 

V= 27t p fVsin ddrdd = — [4(cos 2df sm 6 



dd. 



Transforming, by putting x = cos 6, since 
cos 26 =z 2 cos 2 6 — 1, 

V= — [ (2x 2 - iWx; 
3 Jvi 

and again, by putting x \/2 = sec 0, 

7r 7Ttf 3 4/2 f L d^ ^ _/,*. 7T^ 3 V2f; Sin 4 0,^ 

F = ?— 4 tan 4 sec <pdcf>= — 4 — =-^0. 

3 Jo 3 JoCOS 5 

Using now the formula of reduction (6), Art. 70, we find 

[-- 
Jc< 



in 4 _ sin 3 3 sin 3. 1 -f- sin a 
:0s 5 4 cos 4 8 cos 2 8 cos 



whence, applying the limits, 



Polar Coordinates in Space, 

138. The double integral employed in the preceding arti- 
cle is in fact an application of the polar system of coordinates 
in space. In this system, a point is determined by polar coor- 
dinates in a plane which passes through a fixed axis and makes 
the diedral angle with a fixed plane of reference passing 
through the axis. In Art. 136, the fixed axis is the initial line 



§XI] 



POLAR COORDINATES IN SPACE. 



169 



OA, and taking the plane of the paper as the plane of refer- 
ence, the third coordinate is the angle described by the 
upper half-plane in the revolution of the figure. The differen- 
tial of the motion of P when varies is r sin 6 dcp, and this 
motion is at right angles to the plane of the differentials dr and 
rdd\ hence the ultimate element of volume is r 2 sin 6 dr d6 dcp. 
But in the case of the solid of revolution the limits of the (^-in- 
tegration are O and 2n, so that the result of this integration is 
2n r 2 sin ddrdd, the element of the double integral derived in 
Art. 136. 

139. In the equations of transformation connecting these 
coordinates with the rectangular system, it is usual to take the 
axis of z as that about which the angle 
is described, and the plane xz as that 
for which <p = o. Then in the plane 
ZOR, Fig. 24, the line OR is usually 
taken as the initial line for the polar 
coordinates p and #, so that the three 
coordinates of P are 

p=OP, 6 = P0R, <p = AOR. 
Thi.s p in this system is the distance Fig. 24. 

of the point P from a fixed pole, and 6 and correspond to the 
spherical coordinates latitude and longitude on a sphere whose 
radius is p. Denote the radius of the circle of latitude BP by 
r ; then 

r — p cos 6, 

and r and are the polar coordinate? of the projection of P in 
the plane of xy. Hence the formulae connecting the rectangu- 
lar coordinates x, y, z with p, 6 and are 




x = r cos = p cos 6 cos 0, 
y = r sin = p cos 6 sin 0, 
z = p sin 6. 



I70 GEOMETRICAL APPLICATIONS. [Art. 1 39. 

The differentials of the motions of Pwhen one of the coor- 
dinates p, 6, and varies, the other two remaining constant, 
are, respectively, 

dp, pdB, and p cos 6 d(p; 

hence the element of volume is their product, 

p 2 cos 6 dp d6 d<p, 

in which the factor cos 6 occurs, in place of the factor sin 6 which 
appears in the element derived in the preceding article, because 
the axis of rotation is now perpendicular to the initial line. 

In the case of a sphere whose centre is at the pole, all the 
limits are constant, and we have 



Volumes in general. 

140. We have seen in Art. 123 that the boundary of the 
area expressed by a double integral may consist in part of lines 
whose equations contain only one of the variables, namely, 
chat for which the final integration takes place. But, as ex- 
plained in Art. 125, in the general case, it is necessary to regard 
fhe given area as made up of positive or negative parts of the 
kind just mentioned. This is done by drawing the loci of con- 
stant values of the final variable through the intersections of 
the lines forming the boundary, or else tangent to them as in 
Fig. 19. These parts are then expressed by separate integrals. 

So also we have seen in Art. 126 that the volume expressed 
by a triple integral may be bounded in part by a surface whose 
equation contains only two of the variables, namely, those for 
which the last two integrations take place. But, in the general 
case, it is necessary to separate the volume into parts, by means 



§ XL] VOLUMES IN GENERAL. \J\ 

of such surfaces passed through the lines of intersections of the 
bounding surfaces, or edges of the given solid. 

141. Figs. 19 and 20 illustrate this for rectangular coordi- 
nates, but similar considerations apply to any other system, and 
enable us to decide whether it is possible to express a given 
volume by a single integral. For example, let it be required 
to find the volume common to the solid of revolution produced 
by the half-cardioid OAB, Fig. 25, revolving about its axis OB y 
and the sphere whose centre is at the pole and whose radius 
is OC = c. The volume is the result of integrating the element 

271T 1 sin 6 dr dd 

(found as in Art. 136) over the area OAC. For this area of 
integration, 6 = 7t and r = o give one constant limit for each 
variable, and the others are determined by the equation of 

the arc OA of the cardioid 

r — a{\ — cos 6). . . (1) 

and that of the circular arc AC, 

r = c. ... (2) 

Since equation (1) contains both variables, while equation (2) 
contains r only, we can, by integrating first for (and using 
equation (1) for one of its limits), express the required magni- 
tude by a single integral ; thus, 

V '= 2/t| I r 2 sin 6 dO dr — 2n\ 7^(1 + cos 6)dr. 
Substituting the value of cos 6 from equation (1), 

V= 27T [ ' A2 - -W = ^O - IC\ 




17 2 GEOMETRICAL APPLICATION'S. [Art. 141. 

If in this example we integrate first for r, it becomes neces- 
sary to find 6 lt the value of for the point of intersection A, 
and, regarding the area of integration as separated into two 
parts by the radius vector OA, to form two integrals, in one of 
which the upper limit for r is taken from equation (1), and the 
limits for 6 are o* and 6 1 ; while in the other r is taken from 
equation (2), and the limits for are 6 l and n. 



Examples XI. 

1. The paraboloid of revolution 

x 2 + y = cz 

is pierced by the right circular cylinder 

x 2 + y = ax, 

whose diameter is a, and whose surface contains the axis of the 
paraboloid; find the volume between the plane of xy and the sur- 
faces of the paraboloid and of the cylinder. $na* 

3 2 ' ' 

2. Find the volume cut from a sphere whose radius- is a by the 
cylinder whose base is the curve 

a 2a 7t 8a 
r = a cos 30. . 

3 9 

3. Find the volume cut from a sphere whose radius is a by the 
cylinder whose base is the curve 

r* = 3 <cos a + F sin 2 fl, 

1 ^ 4"*?* *6, 2 72X 3 
supposing < a. — - (a — by. 

*This limit also is in fact determined by the intersection of two sides of the 
area of integration, namely, that of the curve (1) and r — o the vanishing inner 
edge of the area. 



§ XL] EXAMPLES. 1/3 

4. A right cone, the radius of whose base is a and whose alti- 
tude is &, is pierced by a cylinder whose base is a circle having for 
diameter a radius of the base of the cone; find the volume common 
to the cone and the cylinder. ba . ,. 

¥ (9* - .6)- 

5. The axis of a right cone whose semi-vertical angle is a coin- 
cides with a diameter of the sphere whose radius is a, the vertex 
being on the surface of the sphere ; find the volume of the portion 
of the sphere which is outside of the cone. /^na cos 4 a 

3 

6. Find the volume produced by the revolution of the lemniscata 

22 n 

r = a cos 2C7 

.. . ...... 71*0*^2 

about a perpendicular to the initial line. — . 

8 

7. Find the volumes generated by the revolution of the large 
loop and by one of the small loops of the curve 

r = a cos cos 26 

about a perpendicular to the initial line. 

tc a 7za 7t a Tta 
16 5 ' 32 10* 

8. What is the volume of the cavity which will just permit the 
cardioid 

r == a{i — cos 6) 

to rotate about a line on its plane perpendicular to the initial line ? 

i6_+_55 , 

7ta . 

4 

9. Find the whole volume enclosed by the surface 

(x> +/ + *»)• = a'xyz. 

Transform to the coordinates p, 0, 0, and show that the solid con- 
sists of four equal detached parts. a s 

6" 



174 GEOMETRICAL APPLICATIONS. [Ex. XI. 

10. Find the volume of that part of the sphere 
x* + f 4- z % = a(x + y 4- z) 

for which all three coordinates are positive. 27t + I » 

4 

ii. Find the volume between the surface generated by the 
revolution of the cardioid 

r = a(i — cos 6) 

about the initial line and the plane which touches this surface in a 
circle. na* 

192' 



XII. 
Rectification of Plane Curves. 

(42. A curve is said to be rectified when its length is deter- 
mined, the unit of measure to which it is referred being a 
right line. 

It is shown in Diff. Calc, Art. 314 [Abridged Ed., Art. 164], 
that, if s denotes the length of the arc of a curve given in 
rectangular coordinates, we shall have 

ds = V(dx* + df). 

If the abscissas of the extremities of the arc are known, s is 
found by substituting for dy in this expression its value in 
terms of x and dx, and integrating the result between the 
given values of x as limits. Thus, to express the arc measured 
from the vertex of the semi-cubical parabola 

ajfi= x 3 



S xii.] 



RECTIFICATION OF PLANE CURVES. 



J 75 



in terms of the abscissa of its other extremity, we derive, from 
the equation of the curve, 



dy — 



3 Vxdx 

2V a 



whence 
Integrating, 



2 Va 



= \/{gx + ao) dx 

2 \/a J o 



-(gx + 4a) 2 

27 Va xy ^' 27 



143. When x and j are most conveniently expressed as 
functions of a third variable, the expression for ds in terms of 
this variable may be used. For 
example, in the case of the curve 



$Aif 



= I 




we put, as in Art. 107, 
x = a sin 3 if>, 
y = b cos 3 ip ; 
whence 

dx = $a sin 2 ip cos ip dip, 
dy = — lb cos 2 ip sin ip dip ; 
and therefore 

ds = 3 sin ^ cos ^ dip\/(d z sin 2 ^ + ^cos 2 ^) 
= M(^ 2 - ^) sin 2 '/' + £ 2 ] ^(sin 2 ^). 



I76 GEOMETRICAL APPLICATIONS. [Art. 143. 

The points A and B, Fig. 26, correspond to tp — o and 
tp = \-n respectively; hence, integrating, we have 

a z? _ ( a * sin2 * + ^ cos' 0)r^ _ a 3 - £ 3 _ a 2 + ^ + ff 
arc AL> — 







Change of the Sign of ds. 

144. In the example above, x and 7 being one-valued func- 
tions of tp, every value of tp corresponds to a definite point of 
the curve; and, as tp varies from o to 27T, the point (x, y) de- 
scribes the whole curve in the direction ABCD. But it will be 
noticed that, dtp remaining positive, the value of ds becomes 
zero and changes sign when tp passes through either of the val- 
ues o, %7t, 7T, or |tt. This corresponds geometrically to the fact 
that the point (x, y) stops and reverses the direction of its mo- 
tion, forming a stationary point or cusp at each of the points A, 
B, C, and D, as shown in Fig. 26. Such points are thus indi- 
cated by a change of sign in ds, and the arcs between them 
must be considered separately, because the corresponding def- 
inite integrals have different signs. 



Polar Coordinates. 

I45. It is proved in Diff. Calc, Art. 317 [Abridged Ed., 
Art. 167], that when the curve is given in polar coordinates 

ds = V(dr> + r 2 dP). 

This is usually expressed in terms of 6. For example, the 
equation of the cardioid is 

r = a (1 — cos 0) = 2a sin 2 ^#; 
whence . dr == 2a sin \d cos \6 d6, 

and by substitution 

ds = 2a sin \d dO. 



§ XII.] RECTIFICATION OF CURVES. 1 77 

The limits for the whole perimeter of the curve are o and 27T, 
and ds remains positive for the whole interval. Therefore 

f 27r 6 #~| 27r 

s — 2a\ sin — dd = — da cos - = 8a. 
Jo 2 2_| 



Rectification of Curves of Double Curvature. 

146. Let a denote the length of the arc of a curve of double 
curvature ; that is, one which does not lie in a plane, and sup- 
pose the curve to be referred to rectangular coordinates ;r, j/ 
and z. If at any point of the curve the differentials of the 
coordinates be drawn in the directions of their respective axes, 
a rectangular parallelopiped will be formed, whose sides are 
dx, dy and dz, and whose diagonal is da. Hence 

da = V(dx 2 + df + dz 2 ). 

The curve is determined by means of two equations connect- 
ing x, y and z, one of which usually expresses the value of y in 
terms of x, and the other that of z in terms of x. We can 
then express da in terms of x and dx. 

If the given equations contain all the variables, equations 
of the required form may be obtained by elimination. 

147. An equation containing the two variables x and y 
only is evidently the equation of the projection upon the plane 
of xy of a curve traced upon the surface determined by the 
other equation. Let s denote the length of this projection ; 
then, since ds 2 = dx 2 + dj?, 

da = V(ds> + dz 2 ), 

in which ds may, if convenient, be expressed in polar coordin- 
ates ; thus, 

da= Vidr 2 + r 2 dd 2 + dz 2 ). 



1/8 GEOMETRICAL APPLICATIONS. [Art. 148. 

!48. As an illustration, let us use this formula to deter- 
mine the length of the loxodromic curve from the equation of 
the sphere, 

x* +f + z* = a \ (1) 

upon which it is traced, and its projection upon the plane of 
the equator, of which the equation is 

2a = V{X 2 + /)(Vtan-i^ + £ -«tan-i|\ ^ 

or in polar coordinates 

2a = r (£ nB + e- n9 ) (2) 

Equation (1) is equivalent to 

r 2 + £ = a 2 ; 

and, denoting the latitude of the projected point by <f>, this 

gives 

2 = a sin (f>, r = a cos $..-.. (3) 

In order to express dd in terms of (j), we substitute the value 
of r in (2) ; whence 

£ nB _l_ € - m9 — 2 seC 0, (4) 

and by differentiation 

£ n6 _ s -nQ — _ gec ^ tan ^ (cj\ 

n ciu 

Squaring and subtracting equation (5) from equation (4), 



4 sec 2 



2 * 2^1 



which reduces to 



sec 2 



§ XII.] LENGTH OF THE LOXODROMIC CURVE. 1 79 



From equations (3) and (6) 




r i d& 2 = - 9 d(f?, 




df 2, = a 2 sin 2 (f) d(/?, 




dz 1 = a 2 cos 2 <j) d(p ; 




whence substituting in the value of dff (p. 


177) 



da = a V ( 1 + -2 ) d<j>. 



Integrating, 

cr — a-^ d$ = a — ^ ' (/? — a), 

n J a n 

where a and /3 denote the latitudes of the extremities of 
the arc. 

Examples XII. 

1. Find the length of an arc measured from the vertex of the 
catenary 

X X 



and show that the area between the coordinate axes and any arc is 
proportional to the arc. * 



=H- 7 -n 



A = cs. 

2. Find the length of an arc measured from the vertex of the 
parabola 

\/{ax + x ) + a log -j . 



1 80 GEOMETRICAL APPLICATIONS. [Ex. XI L 



3. Find the length of the curve 

e* + 1 



ty 



between the points whose abscissas are a and b. 



lo z-^rzn + a -t* 



4. Find the length, measured from the origin, of the curve 

2 2 
. a — x 
y = a log g — - . 



. a + x 

a log — — x. 

a — x 



5. Given the differential equation of the tractrix, 

dy _ y 



dx *V-/)' 

and, assuming (o, a) to be a point of the curve, find the value of s as 
measured from this point, and also the value of x in terms of y ; that 
is, find the rectangular equation of the curve. 

y 

s = a log — . 
x — a log — — — — — V(a — y). 

• y 

6. Find the length of one branch of the cycloid 

x — a (ip — sin ip), y = a (1 — cos ip). 

Sa. 

7. When the cycloid is referred to its vertex, the equations being 

x — a (1 — cos tp), y = a{ip -{■ sin ip), 

prove that s = \/(8ax). 



§ XII.] EXAMPLES. 16 1 

8. Find the length from the point (a, o) of the curve 

x = 2a cos ip — a cos 2^?, 

j> = 2# sin ip — a sin 2ip. 

8a (1 — cos£^). 

9. Show that the curve, 

x = 3 # cos ^ — 2# cos 3 ^>, y = 2a sin 3 ^, 

has cusps at the points given by ip = o and ip = n ; and find the 
whole length of the curve. 12a. 

10. Find the length of the arc of the parabola 

between the points where it touches the axes. 

a* + d s a'F [V(a 2 4- F) + a] jy(a a + F) + t] 

a* + b l + \<? + jP)« l0g ^ 

11. Show that the curve 

x = 2a cos 2 6 (3 — 2 cos 2 0), ^ = 4^ sin 6 cos 3 

has three cusps, and that the length of each branch is — . 

3 

12. Find the length of the arc between the points at which the 
curve 

x = a cos 2 #coS2#, y = tf sin 2 # sin 26 

2 + V2 
cuts the axes. a. 



1 82 GEOMETRICAL APPLICATIONS. [Ex. XII. 

13. Show that the curve 

p 

x —. a cos ip (1 + sin 2 ?/?), 

y — a sin ip cos 2 ip 

is symmetrical to the axes, and find the length of the arcs between 

the cusps. / ' . 1 

0(4/2— sm- 1 — 
\ V3 



14. Find the length of one branch of the epicycloid 



0(4/2 + COS" 

V3> 



x = (a + b) cos ^p — b cos — 7— ^, 



_y = (0 + b) sin ^> — b sin — - — 



Sb (0 + J) 



15. Show that the curve 



x = 90 sin ip — 40 sin 3 ^, 
j = — 30 cos ^ + 40 cos 3 ip 

is symmetrical to the axes, and has double points and cusps : find the 
lengths of the arcs, {a) between the double points, (/3) between a 
double point and a cusp, and (y) the arc connecting two cusps, and not 
passing through the double points. 

(a), a {n + 3 4/3) ; 

(y)> a (3V3- *)> 
16. Find the whole length of the curve 

x = 30 sin i/j — a sin*tp, 

y=. a cos 3 */\ 3^0. 



§ XII ] EXAMPLES. 183 

17. Find the length, measured from the pole, of any arc of the 
equiangular spiral 

r — ae n0 , 
in which n = cot a. r sec a. 

18. Prove by integration that the arc subtending the angle e at the 
circumference in a circle whose radius is a, is 2ah. 

19. Find the length, measured from the origin, of the curve defined 
by the equations 

x x 3 



y ~ 2a 1 Z ~6a 2 - 



X+ 6a>- 



20. Find the length, measured from the origin, of the intersection of 
the surfaces 

y = 471 sin x, z= 211" (2X + sin 2x). 

(4« 2 + l)x + 272 2 Sin 2X. 

21. Find the length, measured from the origin, of the intersection of 
the cylindrical surfaces 

(y — x) 2 = 4ax, ga (z — x) 2 = 4X 3 . 

2X\ ., v , 

+ 2 y(ax)+x. 



22. If upon the hyperbolic cylinder 



y z 



? = x - 



a curve whose projection upon the plane of xy is the catenary 

X X 



2 



be traced, prove that any arc of the curve bears to the corresponding 
arc of its projection the constant ratio V{b 2 + c 2 ) : c. 



1 84 



GE OME TRICA L A P PLICA TIONS. 



[Art. 149 



XIII. 

Surfaces of Solids of Revolution. 

149' The surface of a solid of revolution may be generated 
by the circumference of the circular section made by a plane 

perpendicular to the axis of revolu- 
tion. Thus in Fig. 27, the surface 
produced by the revolution of the 
curve AB about the axis of x is re- 
garded as generated by the circum- 
ference PQ. The radius of this cir- 
cumference is y, and its plane has a 
motion whose differential is dx, but 
every point in the circumference itself 
has a motion whose differential is ds } s 
denoting an arc of the curve AB. 
Hence, denoting the required surface by S, we have 

dS = 2ny ds = 27ty y ' (dx 2 + df). 

The value of dS must of course be expressed in terms of a single 
variable before integration. 

150. For example, let us determine the area of the zone of 
spherical surface included between any two parallel planes. 
The radius of the sphere being a, the equation of the revolv- 
ing curve is 

x 2 4- j*/ 2 = a 2 ; 

whence y — V(a 2 — x 2 ), 

x dx 
dy ~ ~ ~V(a 2 - x 2 ) ' 
adx 




and 



V{a 2 — x 2 ) 
dS = 2na dx ; 



§ XIII.] SURFACES OF SOLIDS OF REVOLUTION. 



therefore 



5 = 2na dx — 2na (x 2 — x x ) 



Since x 2 — x 1 is the distance between the parallel planes, 
the area of a zone is the product of its altitude by 27ta, the 
circumference of a great circle, and the area of the whole sur- 
face of the sphere is 4x0?. 

151. When the curve is given in polar coordinates, it is con- 
venient to transform the expression for S to polar coordinates. 
Thus, if the curve revolves about the initial line, 

S=27r\yds=27tlr sin 6 \(dr* + r> d&). 
For example, if the curve is the cardioid 



r —2a sin* — 

2 



we find, as in Art. 145, 



Hence 



ds = 2a sin — 6 dd. 

2 



s = 1671a 2 ("sin 4 - e cos - e dd 

Jo 2 2 



= ^ sin 5 - 6 

5 2 Jo 



Areas of Surfaces in General. 

152. Let a surface be referred to rectangular coordinates x, 
y and z ; the projection of a given portion of the surface upon 
the plane of xy is a plane area determined by a given relation 
between x and y. We may take as the elements of the surface 
the portions which are projected upon the corresponding 



1 86 



GE OME TRICA L A P PLICA TIONS. 



[Art. 152. 



elements of area in the plane of xy. If at a point within the 
element of surface, which is projected upon a given element 
Ax Ay, a tangent plane be passed, and if y denote the inclina- 
tion of this plane to the plane of xy,the area of the correspond- 
ing element in the tangent plane is 

sec y Ax Ay. 

The surface is evidently the limit of the sum of the elements 
in the tangent planes when Ax and Ay are indefinitely dimin- 
ished. Now sec y is a function of the coordinates of the point 
of contact of the tangent plane ; and since these coordinates 
are values of x and y which lie respectively between x and 
x + Ax and between y and y -f Ay, it follows, as in Art. 122, 
that this limit is 



S = sec y dx dy. 



153. The value of sec y may be derived by the following 

method. Through the point P of 
the surface let planes be passed 
parallel to the coordinate planes, 
and let PD, and PE, Fig. 28, be the 
intersections of the tangent plane 
with the planes parallel to the 
planes of xz andyz. Then PD and 
PE are tangents at P to the sec- 
tions of the surface made by these 
FlG 2g planes. The equations of these 

sections are found by regarding y 
and x in turn as constants in the equation of the surface ; there- 
fore denoting the inclinations of these tangent lines to the plane 
of xy by $ and ip, we have 




. dz 

tan = — 

dx 



and 



tariff) = 



dz 
dy' 



§ XIIL] AREAS OF SURFACES IN GENERAL. 1 87 

dz dz 
in which — and -7- are partial derivatives derived from the equa- 
dx ay ^ 

tion of the surface. 

If the planes be intersected by a spherical surface whose 
centre is P> ABE is a spherical triangle right angled at A, 
whose sides are the complements of <f> and ip. Moreover, if a 
plane perpendicular to the tangent plane PED be passed 
through AP, the angle FPG will be y, and the perpendicular 
from the right angle to the base of the triangle the comple- 
ment of y. 

Denoting the angle EAF by 6, the formulae for solving 
spherical right triangles give 

n tan ip , . „ tan <f> 

cos # = -, and sin = . 

tan y tan y 

Squaring and adding, 

_ tan 2 if) + tan 2 $ 
1 " SmV ' 

or tan 2 y = tan 2 ip + tan 2 $ ; 
whence sec* y = I + (g)' + (|)\ 

Substituting in the formula derived in Art. (152), we have 

MM- ♦©■♦<£)>* 

154. It is sometimes more convenient to employ the polar 



1 88 GEOMETRICAL APPLICATIONS. [Art. 1 54. 

element of the projected area. Thus the formula becomes 
5 = sec yr dr dd, 

where sec y has the same meaning as before. 

For example, let it be required to find the area of the sur- 
face of a hemisphere intercepted by a right cylinder having a 
radius of the hemisphere for one of its diameters. From the 
equation of the sphere, 

x*+f+2*=O i 9 ...... . (1) 



we derive 


dz 
dx~~ 


X 

~z y 




dz 
dy~ 


y . 

' z ' 


whence 














sec^= |/ 1 


+ 


(£)• 


► (!)']■ 


a 

~z ; 


therefore 






S = a[ 


irdrdS 





the integration extending over the area of the circle 

r — a cos (2) 

Since equation (1) is equivalent to 

z 2 + 7* = a\ 



§ XIII.] AREAS OF SURFACES IN GENERAL. 1 89 

From (2) the limits for r are r x = 0, and r 2 = a cos #, 
hence 



S = a 2 [(i -sm6)dd, 



in which a sin 6 is put for the positive quantity V{a 2 — r.?). The 
limits for 6 are — \n and \n, but since sin 6 is in this case to 
be regarded as invariable in sign, we must write 



5 = 2a 2 2 (1 — sin 6) dd = na 2 — 2a 2 . 

Jo 



If another cylinder be constructed, having the opposite radius 
of the hemisphere for diameter, the surface removed is 
2na 2 — 4a 2 , and the surface which remains is 4a 2 , a quantity 
commensurable with the square of the radius. This problem 
was proposed in 1692, in the form of an enigma, by Viviani, a 
Florentine mathematician. 



Examples XIII. 

1. Find the surface of the paraboloid whose altitude is a, and the 
radius of whose base is b. 



2. Prove that the surface generated by the arc of the catenary given 
in Ex. XII., 1, revolving about the axis of x, is equal to 

n{cx + sy). 

3. Find the whole surface of the oblate spheroid produced by th e 



I90 GEOMETRICAL APPLICATIONS. [Ex. XIII. 



revolution of an ellipse about its minor axis, a denoting the major, 
b the minor semi-axis, and e the excentncity, 



a 



1>\ 1 + e 

27ta + 7t - log . 

e ° 1 — e 

4. Find the whole surface of the prolate spheroid produced by the 
revolution of the ellipse about its major axis, using the same notation 
as in Ex. 3. 

72 , 7 sin -1 ^ 

2710 + 271 ab — , 

e 

5. Find the surface generated by the cycloid 

x — a {ip — sin ip), y = a (1 — cos ip) 



revolving about its base. — na 

3 



6. Find the surface generated when the cycloid revolves about the 
tangent at its vertex. 



7. Find the surface generated when the cycloid revolves about its 



axis. 



%7ld 



(-!) 



8. Find the surface generated by the revolution of one branch of 
the tractrix (see Ex. Xli , 5) about its asymptote- 



§ XIII] EXAMPLES. L9 [ 

9. Find the surface generated by the revolution about the axis of 
x of the portion of the curve 

which is on the left of the axis of y. 

7r[+/2 + log (1 + \2)~\. 

10. Find the surface generated by the revolution about the axis of 
x of the arc between the points for which x = a and x = b in the 
hyperbola 

xy — k 2 . 



7t# 



' b 2 + Vjk* + b A ) V{fr + a') _ V(k* + b*) "I 
g a 2 4- f(>& 4 + a) + « a ^ 2 



11. Show that the surface of a cylinder whose generating lines are 
parallel to the axis of z is represented by the integral 



S 



= \z ds, 



where s denotes the arc of the base in the plane of xy. Hence, 
deduce the surface cut from a right circular cylinder whose radius is 
a, by a plane passing through the centre and making the angle a with 
the plane of the base. 2a? tan a. 

12. Find the surface of that portion of the cylinder in the problem 
solved in Art. 1 54, which is within the hemisphere. 2a*. 

13. Find the surface of a circular spindle, a being the radius and 
2c the chord. 

47ta\ c — 4/(0' 



c — V{a* ~ ^sin- 1 - . 



I9 2 GEOMETRICAL APPLICATIONS. [Art. 1 55. 



XIV. 

The Area generated by a Straight Line moving in any 
Manner in a Pla7ie. 

155. If a straight line of indefinite length moves in any man- 
ner whatever in a plane, there is at each instant a point of the 
line about which it may be regarded as rotating. This point we 
shall call the centre of rotation for the instant. The rate of 
motion of every point of the line in a direction perpendic- 
ular to the line itself is at the instant the same as it would 
be if the line were rotating at the same angular rate about this 
point as a fixed centre.* Hence it follows that the area 
generated by a definite portion of the line has at the instant 
the same rate as if the line were rotating about a fixed instead 
of a variable centre. 

156. Suppose at first that the centre of rotation is on the 
generating line produced, p 1 and p 2 denoting the distances from 
the centre of the extremities of the generating line, and let (j) 
denote its inclination to a fixed line. By substitution in the 
general formula derived in Art. 1 10, we have 

dA = l -(p?-p 1 2 )d<f>. 



* Compare Diff. Calc, Art. 332 [Abridged Ed., Art. 176J, where the moving 
line is the normal to a given curve, and the centre of rotation is the centre of cur- 
vature of the given curve. If the line is moving without change of direction, the 
centre is of course at an infinite distance. 

When the line is regarded as forming a part of a rigidly connected system in 
motion, its centre of rotation is the foot of a perpendicular dropped upon it from 
the instantaneous centre of the motion of the system. Thus, if the tangent and 
normal in the illustration cited are rigidly connected, the centre of curvature, C, is. 
the instantaneous centre of the motion of the system, and the point of contact, P t 
is the centre of rotation for the tangent line. 



§ XIV.] AREA S GENERA TED BY MO VING LINES. T 93 



Applications, 

157. The area between a curve and its evolute may be 
generated by the radius of curvature p, whose inclination to 
the axis of x is <j) + \n, in which (j) denotes the inclination 
of the tangent line. Since the centre of rotation is one 
extremity of the generating linep, the differential of this area 
is found by substituting in the general expression p 1 = o and 
p 2 = p. Hence when p is expressed in terms of <f>, 



=\\f?d<i> 



expresses the area between an arc of a given curve, its evolute, 
and the radii of curvature of its extremities, the limits being 
the values of (f) at the ends of the given arc. 

158. For example, in the case of the cardioid 

r — a{\ — cos 6), 

it is readily shown, from the results obtained in Art. 145, that 
the angle between the tangent and the radius vector is \d', and 
therefore <f> = \Q, and 

ds 4a . 6 

<W 3 3 

To obtain the whole area between the curve and its evolute, 
the limits for 6 are o and 2n ; hence the limits for </> are o 
and 3?r. Therefore 



2J0 9 Jo 3 



3 



159. As another application of the general formula of 
Art. 156, let one end of a line of fixed length a be moved 



T 94 GEOMETRICAL APPLICATIONS. [Art. 1 59. 

along a given line in a horizontal plane, while a weight at* 
tached to the other extremity is drawn over the plane by the 
line, and is therefore always moving in the direction of the 
line itself. The line of fixed length in this case turns about 
the weight as a moving centre of rotation. Hence the area 
generated while the line turns through a given angle is the 
same as that of the corresponding sector of a circle whose 
radius is a. 

The curve described by the weight is called a tractrix, and 
the line along which the other extremity is moved is the direc* 
trix. When the axis of x is the directrix, and the weight 
starts from the point (o, a)> the common tractrix is described ; 
hence the area between this curve and the axes is \na % . 

160. Again, in the generation of the cycloid, DifT. Calc, 
Art. 288 [Abridged Ed., Art. 156], the variable chord RP may 
be regarded as generating the area. The point R has a motion 
in the direction of the tangent RX; the point P partakes of 
this motion, which is the motion of the centre C, and also has 
an equal motion, due to the rotation of the circle in the direc- 
tion of the tangent to the circle at P. Since the tangents 
at P and R are equally inclined to PR, the motion of P in a 
direction perpendicular to PR is double the component, in this 
direction, of the motion of R. Therefore the centre of rota- 
tion of PR is beyond R at a distance from it equal to PR. 
Hence, denoting PRO by ^, 

p t ■— PR = 2a sin ^, p 2 = 2PR = 4a sin <j>. 

Substituting in the formula of Art. 156, we have for the area 
of the cycloid, since PRO varies from o to 7T, 



A = 6a 2 \ sin 2 <frd<f> = 3^. 



§ XIV.] 



SZGX OF THE GENERATED AREA. 



195 




Fig. 29. 



Sign of the Generated Area, 

161. Let AB be the generating line, and C the centre of 
TOtation. The expression, 

dA =i(pi-Pi*)M, (1) 

for the differential of the area, was obtained upon the supposi- 
tion that A and B were on the same side of C. Then suppos- 
ing P2 > Pi, and that the nne rotates in the positive direction 
as in figure 29, the differential of the area is 
positive; and we notice that every point in the 
area generated is swept over by the line 
AB, the left hand side as we face in the 
direction A B preceding. 

162. We shall now show that in every 
case, the formula requires that an area 

swept over with the left side preceding, shall be considered 
as positively generated, and one swept over in the opposite 
direction as negatively generated. 

In the first place, if C is between A and 
B so that Pi is negative, as in figure 30, Pi 2 
is still positive, and formula (1) still gives 
the difference between the areas generated 
by CB and AC. Hence the latter area, 
which is now generated by a part of the 
line AB, must be regarded as generated 
negatively, but the right hand side as we 
face in the direction AB of this portion of the line is now 
preceding, which agrees with the rule given in Art. 161. 

Again, if C is beyond B, the formula gives the difference 
of the generated areas ; but since Pj 2 is numerically greater 
than p}, in this case, dA is negative, and the area generated by 
AB is the difference of the areas, and is negative by the rule. 




Fig. 30. 



196 



GEOMETRICAL APPLICATIONS. 



[Art. 162. 



Finally, if the direction of rotation be reversed, d<j> and 
therefore dA change sign, but the opposite side of each por- 
tion of the line becomes in this case the preceding side. 

163. We may now put the expression for the area in another 
form. For 



dA = ^(pi-p})J4 



(fh 



p,)^- 1 ^; 



whatever be the signs of p 2 and p x , the first factor is the length 
of AB, which we shall denote by /, and the second factor is 
the distance of the middle point of AB from the centre of 
rotation, which we shall denote by p m . Hence, putting 



P2- Pi = I, 
we have 



and 



9% + A _ 



A = \lp m d<j>. 



(2) 



Since p m d<j> is the differential of the motion of the middle point 
in a direction perpendicular to AB, this expression shows that 
the differential of the area is the product of this differential by 
the length of the generating line. 



Areas generated by Lines whose Extremities describe 
Closed Circuits. 



^< 



164. Let us now suppose the generating line AB to move 
from a given position, and to return to the 
same position, each of the extremities A and 
B describing a closed curve in the positive 
direction, as indicated by the arrows in figure 
31. It is readily seen that every point which 
is in the area described by B, and not in that 
described by A, will be swept over at least 
once by the line AB, the left side preceding, 

Fig. 31. and if passed over more than once, there will be 




§ XIV.] AREAS GENERATED BY MOVING LINES. 1 97 

an excess of one passage, the left side preceding. Therefore 
the area within the curve described by B, and not within that 
described by A, will be generated positively. In like manner 
the area within the curve described by A, and not within that 
described by B, will be generated negatively. Furthermore, all 
points within both or neither of these curves are passed over, 
if at all, an equal number of times in each direction, so that the 
area common to the two curves and exterior to both disap- 
pears from the expression for the area generated by AB. 

Hence it follows that, regarding a closed area whose perimeter 
is described in the positive direction as positive, the area generated 
by a line returning to its original position is the difference of the 
areas described by its extremities. This theorem is evidently 
true generally, if areas described in the opposite direction are 
regarded as negative. 



Amslers Planimeter. 

165. The theorem established in the preceding article may 
be used to demonstrate the correctness of the method by 
which an area is measured by means of the Polar Planimeter, 
invented by Professor Amsler, of Schaffhausen. 

This instrument consists of two bars, OA and AB, Fig. 32, 
jointed together at A. The rod OA turns on 
a fixed pivot at O, while a tracer at B is carried 
in the positive direction completely around 
the perimeter of the area to be measured. At 
some point C of the bar AB a small wheel is 
fixed, having its axis parallel to AB, and its 
circumference resting upon the paper. When 
B is moved, this wheel has a sliding and a roll- 
ing motion ; the latter motion is recorded by 
an attachment by means of which the number Fig. 32. 

of turns and parts of a turn of the wheel are registered. 




I9 8 GEOMETRICAL APPLICATIONS. [Art. l66„ 

166. Let M be the middle point of AB, and let 

OA =a, AB = b, MC=c. 

Since b is constant, the area described by AB is by equation (2), 
Art. 163, 

Area AB = b\p m d(f> (1) 

Denoting the linear distance registered on the circumference 
of the wheel by s, ds is the differential of the motion of the 
point C, in a direction perpendicular to AB, and since the dis- 
tance of this point from the centre of rotation is p m + c, 

ds = (p m + c) d(j)\ 
substituting in (1) the value of p m d$, 

Area AB = b ids - be id<f> (2) 

167. Two cases arise in the use of the instrument. When,, 
as represented in Fig. 32, O is outside the area to be meas- 
ured, the point A describes no area, and by the theorem of 
Art. 164, equation (2) represents simply the area described 

by B. In this case ^ returns to its original value, hence d$ 

vanishes, and denoting the area to be measured by A, equation 
(2) becomes 

A=bs (3) 

In the second case, when O is within the curve traced by B, 
the point A describes a circle whose area is nc?, and the limit- 



§ XIV.] AMSLER'S PLANIMETER. J 99 

ing values of <f> differ by a complete revolution. Hence in this 
case equation (2) becomes 

A — Tta 2 — bs — 27tbc y 

or A = bs + n (a 2 — 2bc)* (4) 

In another form of the planimeter the point A moves in a 
straight line, and the same demonstration shows that the area 
is always equal to bs. 

Examples XIV. 

1. The involute of a circle whose radius is a is drawn, and a tangent 
is drawn at the opposite end of the diameter which passes through the 
cusp ; find the area between the tangent and the involute. 

a*n (3 4- 7t*) 



2. Two radii vectoresof a closed oval are drawn from a fixed point 
within, one of which is parallel to the tangent at the extremity of the 
other ; if the parallelogram be completed, the area of the locus of its 
vertex is double the area of the given oval. 

3. Show that the area of the locus of the middle point of the chord 
joining the extremities of the radii vectores in Ex. 2, is one half the 
area of the given oval. 



* The planimeter is usually so constructed that the positive direction of rotation 
is with the hands of a watch. The bar b is adjustable, but the distance A C is fixed 
so that c varies with b. Denoting A C by q, we have c = q — \b, and the constant 
to be added becomes C = it (a 2 — ibq -+■ b 2 ) in which a and q are fixed and b adjusta- 
ble. In some instruments q is negative. 

It is to be noticed that in the second case s may be negative ; the area is then 
the numerical difference between the constant and bs. 



200 GEOMETRICAL APPLICATIONS. [Ex. XIV. 

4. Prove that the difference of the perimeters of two parallel ovals, 
whose distance is b, is 27tb, and that the difference of their areas is 
the product of b and the half sum of their perimeters. 

5. From a fixed point on the circumference of a circle whose radius 
is a a radius vector is drawn, and a distance b is measured from the 
circumference upon the radius vector produced ; the extremity of b 
therefore describes a limacon : show that the area generated by b when 
b > 2a is the area of the limacon diminished by twice the area of the 
circle, and thence determine the area of the limacon. 

71(20* + b 2 ). 

6. Verify equation (4), Art. 167, when the tracer describes the 
circle whose radius is a + b. 

7. Verify the value of the constant in equation (4), Art. 167, by 
determining the circle which may be described by the tracer without 
motion of the wheel. 

8. If, in the motion of a crank and connecting rod (the line of motion 
of the piston passing through the centre of the crank), Amsler's record- 
ing wheel be attached to the connecting rod at the piston end, deter- 
mine s geometrically, and verify by means of the area described by the 
other end of the rod. 

9. The length of the crank in Ex. 8 being a, and that of the con- 
necting rod b, find the area of the locus of a point on the connecting 
rod at a distance c from the piston end. 



10. If a line AB of fixed length move in a plane, returning to its 
original position without making a complete revolution, denoting the areas 
of the curves described by its extremities by (A) and (B), determine 
the area of the curve described by a ' point cutting AB in the ratio 
m : n. 

n(A) + m{B) 



§ XIV.] 



EXAMPLES. 



201 



11. If the line in Ex. 10 return to its original position after making a 
complete revolution, prove ITolditch's Theorem ; namely, that the area of 
the curve described by a point at the distance c and c from A and B is 



c\A) + c(B) _ 



c + c 



7tCC 



12. Show by means of Ex. 11 that, if a chord of fixed length move 
around an oval, and a curve be described by a point at the distances 
c and c from its ends, the area between the curves will be ncc '. 



XV. 

Approximate Expressions for Areas and Volumes. 

168. When the equation of a curve is unknown, the area 
between the curve, the axis of x y and 
two ordinates may be approximately ex- 
pressed in terms of the base and a lim- 
ited number of ordinates, which are sup- 
posed to have been measured. 

Let ABCDE be the area to be de- 
termined ; denote the length of the base 
by 2J1 ; and let the ordinates at the ex- 
tremities and middle point of the base 
be measured and denoted by y x ,y 2 , and j/ 3 . Taking the base for 
the axis of x, and the middle point as origin, let it be assumed 
that the curve has an equation of the form 

y = A+Bx+C*? + D**; (1) 

then the area required is 

A=f ydx=Ax+2£+ — +— ]* =-(6A+2C&), .(2) 

in which which A and C are unknown. 




Fig. 33. 



202 



GEOMETRICAL APPLICATIONS. 



[Art. 168. 



In order to express the area in terms of the measured 
ordinates, we have from equation (i), 

y x = A + Bh + OP + Dh\ 

y% = A, 

y 3 = A-Bh + C&-DJP; 

whence we derive 

y\ +n = zA + 2C&, 

y\ + 4/2 + y* = 6A + 2C/1 2 ; 

and substituting in (2), 

It will be noticed that this formula gives a perfectly ac- 
curate result when the curve is really a parabolic curve of the 
third or a lower degree. 

169. If the base be divided into three equal intervals, each 
denoted by h, and the ordinates at the extremities and at the 
points of division measured, we have, by assuming the same 
equation, 

- h 

A=\ 2 3 ^dx=^(4A + 3 C/ti) . .... (1) 



From the equation of the curve, 

y t =A 




Fro. 34. 



8 ' 



y% 



-'A 



2 




t- = — 
4 




Bh 


+ 


Ch* 


Dh* 


2 


4 


8 



. Bh Ck* Dk 3 



8 



XV.] SIMPSON'S RULES. 20$ 



whence y x + y± — 2A + 

y% + y* — 2A + 

From these equations we obtain 



9C& 

2 

Of 
2 



A — — ie ' 

and CT= ^-^-^+^ t 

4 

Substituting in equation (1), 

A = ^(j 1 + 3j 2 + 3/3+^). 



Simpson's Rules. 

170. The formulae derived in Articles 168 and 169, although 
they were first given by Cotes and Newton, are usually known 
as Simpsons Rules, the following extensions of the formulae 
having been published in 1743, in his Mathematical Disserta- 
tions. 

If the whole base be divided into an even number n of 
parts, each equal to k, and the ordinates at the points of divis- 
ion be numbered in order from end to end, then by applying 
the first formula to the areas between the alternate ordinates, 
we have 

h 

A = - Cri + 4^2 + 2jj/ 3 + 474 • • • + 4j« + yn+ 1). 

That is to say, the area is equal to the product of the sum of 
the extreme ordinates, four times the sum of the even-nun> 



204 GEOMETRICAL APPLICATIONS. [Art. I/O, 

bered ordinates, and twice the sum of the remaining odd-num- 
bered ordinates, multiplied by one third of the common interval. 
Again, if the base be divided into a number of parts divis- 
ible by three, we have, by applying the formula derived in 
Art. 169, to the areas between the ordinates y t j 4 , y±y n , and so on, 

0/1 

Cotes Method of Approximation, 

171. The method employed in Articles 168 and 169 is 
known as Cotes Method. It consists in assuming the given 
curve to be a parabolic curve of the highest order which can 
be made to pass through the extremities of a series of equi- 
distant measured ordinates. 

The equation of the parabolic curve of the nth. order con- 
tains n + 1 unknown constants ; hence, in order to eliminate 
these constants from the expression for an area defined by the 
curve, it is in general necessary to have n -f 1 equations con- 
necting them with the measured ordinates. Hence, if n de- 
note the number of intervals between measured ordinates over 
which the curve extends, the curve will in general be of the 
nth. degree.* 

* If H denotes the whole base, the first factor is always equivalent to H 
divided by the sum of the coefficients of the ordinates ; for if all the ordinates are 
made equal, the expression must reduce to Hy x . Thus, each of the rules for an 
approximate area, including those derived by repeated applications, as in Art. 170, 
may be regarded as giving an expression for the mean ordinate. The coefficients 
of the ordinates, according to Cotes' method, for all values of n up to n = 10, may 
be found in Bertrand's Calcul Integral, pages 333 and 334. For example (using 
detached coefficients for brevity), we have, when n = 4, 

A = ™ &> 32> I2 ' 32 ' 7 ^ ; 
90 

and when n = 6, 

TT 

A = 57^ [4 J » 2l6 > 2 7> 2 72, 27, 216, 41]. 

040 



§ XV.] THE FIVE-EIGHT RULE. 205 

172. For example, let it be required to determine the area 
between the ordinates y x and y 2t in terms of the three equi- 
distant ordinates y u jj/ 2 and y s , the common interval being h. 

We must assume 

y — A + Bx + Cx*\ 

then taking the origin at the foot of y lf 

,=JW*[. + f + f], 

from which A, B and C must be eliminated by means of the 
equations 

y 2 = A+Bk + Ck 2 , 
y B = A + 2Bh 4- 4Ck\ 

Solving these equations, we obtain 

A =y lf 

2 



If we make a slight modification in the ratios of these last coefficients by sub- 
stituting for each the nearest multiple of 42, we have 

A = - — [42, 210, 42, 252, 42, 210, 42], 
840 

(the denominator remaining unchanged, since the sum of the coefficients is still 
840), which reduces to 

A =— [1, 5, 1, 6, 1, 5, ij. 

This result is known as Weddles Rule for six intervals. The value thus given to 
the mean ordinate is evidently a very close approximation to that resulting from 
Cotes' method, the difference being 

g— fvj + y-, + 15 (y* + ys) — 6 {y% + y 6 ) - aqy 4 ]. 



206 GEOMETRICAL APPLICATIONS. [Art. 1 72. 

and substituting 

-*-— (5*i + 8ft -.?•). 

(73. It is, however, to be noticed, that when the ordinates 
are symmetrically situated with respect to the area, if n is 
even, the parabolic curve may be assumed of the (n + i)th 
degree. For example, in Art. 168, n — 2, but the curve was 
assumed of the third degree. Inasmuch as A, B, C and D 
cannot all be expressed in terms of y lt y 2 , and j/ 3 , we see that a 
variety of parabolic curves of the third degree can be passed 
through the extremities of the measured ordinates, but all of 
these curves have the same area.* 

Application to Solids. 

174. li y denotes the area of the section of a solid perpen- 
dicular to the axis of x, the volume of the solid is y dx, and 

* This circumstance indicates a probable advantage in making n an even num- 
ber when repeated applications of the rules are made. Thus, in the case of six 
intervals, we can make three applications of Simpson's first rule, giving 

TT 

A = —[i, 4, 2, 4, 2, 4, 1], (1) 

or two of Simpson's second rule, giving 

A - ^ [1, 3, 3, 2, 3, 3, 1] (2) 

In the first case, we assume the curve to consist of three arcs of the third degree, 
meeting at the extremities of the ordinates y % andjj/ 5 ; but, since each of these arcs 
contains an undetermined constant, we can assume them to have common tangents 
at the points of meeting. We have therefore a smooth, though not a continuous 
curve. In the second case, we have two arcs of the third degree containing no 
arbitrary constants, and therefore making an angle at the extremity of jy 4 . It is 
probable, therefore, that the smooth curve of the first case will in most cases form a 
better approximation than the broken curve of the second case. 

In confirmation of this conclusion, it will be noticed that the ratios of the 
coefficients in equation (1) are nearer to those of Cotes' coefficients for n = 6, given 
in the preceding foot-note, than are those in equation (2). 



XV.] 



APPLICATION TO SOLIDS, 



20/ 



therefore the approximate rules deduced in the preceding arti- 
cles apply to solids as well as to areas. Indeed, they may be 
applied to the approximate computation of any integral, by 
putting y equal to the coefficient of dx under the integral sign. 
The areas of the sections may of course be computed by 
the approximate rules. 



Woolleys Rule. 

175. When the base of the solid is rectangular, and the 
ordinates of the sections necessary to the application of Simp- 
son's first rule are measured, we may, instead of applying that 
rule, introduce the ordinates directly into the expression for 
the area in the following manner. 

Taking the plane of the base for the plane of xy, and its 
-centre for the origin, let the equation of the upper surface be 
assumed of the form 

z=A +Bx+Cy + Dx 2 + Exy + Ff + Gx* + Hx'*y + Ixf+7f. 

Let 2k and 2k be the dimensions of the base, and denote 
the measured values of z as indicated in 
Fig. 35. The required volume is 



V 



)-h J 



z dy dx. 



This double integral vanishes for every 
term containing an odd power of x or an 
odd power of y : hence 





N ^~~\ 


a q 




|jijX---~^N 


\ 

\ 


a 2 


r^k 


li<h 




\ 


0\ 
a-, 


61 



Fig. 35. 



= —[i2A +4^ + 4^]. 
3 



(I) 



208 GEOMETRICAL APPLICATIONS. [Art. 1 75. 

By substituting the values of x and y in the equation of the 
surface, we readily obtain 

b % = A, (2) 

a x + a % + c x + c 3 = 4A + 4D/1 2 + 4F&, ... (3) 
a 2 -f c 2 + d 1 + b z = 4A + 2Z>^ 2 + 2F&. ... (4) 

From these equations two very simple expressions for the 
volume may be derived ; for, employing (2) and (4), equation 
(1) becomes 

2,hk 
V=— -(#2 + #i + 2^ 2 + b. d + c 2 ); . . . . (4) 

and employing (2) and (3), 

hk 
V— — (>i + a 3 + Sb 2 + c x + c s ) (5) 

Equation (4) is known as Woolleys Rule ; the ordinates employed 
are those at the middles of the sides and at the centre ; in (5), 
they are at the corners and at the centre. 



Examples XV. 

1. Apply Simpson's Rule to the sphere, the hemisphere, and the 
cone, and explain why the results are perfectly accurate. 

2. Apply Simpson's Second Rule to the larger segment of a sphere 
made by a plane bisecting at right angles a radius of the sphere. 

g7ta a 

"TT" 



§ XV.] EXAMPLES. 209 

3. Find by Simpson's Rule the volume of a segment of a sphere, b 
and c being the radii of the bases, and h the altitude. 

^(3^*V+*). 



4. Find by Simpson's Rule the volume of the frustum of a cone, b 
and c being the radii of the bases, and h the altitude. 

— (# + be + <r 2 ). 

5. Compute by Simpson's First and Second Rules, the value of 

f 1 dx 

, the common interval being ^ in each case. 

j 1 ■+■ x 

The first rule gives 0.6931487, and the second rule gives 0.6931505. 

The correct value is obviously log € 2 = 0.6931472. 



6. Find the volume considered in Art. 175, directly by Simpson's 
Rule, and show that the result is consistent with equations (4) and (5). 

hk 
y= — \_a x + a z + d + c 3 4- 4 (a<2 + h +b 3 4- <r 2 ) + 16AJ. 

7. Find, by elimination, from equations (4) and (5), Art. 175, a 
formula which can be used when the centre ordinate is unknown. 

V= — [4(«t + h + h + c 9 ) - (ch + a % + d + <:,)]. 



2IO MECHANICAL APPLICATIONS. [Art. 1 76, 

CHAPTER IV. 

Mechanical Applications. 



XVI. 

Definitions. 

176. We shall give in this chapter a few of the applications 
of the Integral Calculus to mechanical questions. 

The mass or quantity of matter contained in a body is pro- 
portional to its weight. When the masses of all parts of equal 
volume are equal, the body is said to be homogeneous. The 
factor by which it is necessary to multiply the unit of volume 
to produce the unit of mass is called the density, and usually 
denoted by y. 

In the following articles it will be assumed, when not other- 
wise stated, that the body is homogeneous, and that the density 
is equal to unity, so that the unit of mass is identical with the 
unit of volume. When the mass of an area is spoken of, it is 
regarded as a lamina of uniform thickness and density, and the 
unit of mass is taken to correspond with the unit of surface. 
In like manner the unit of mass for a line is taken as identical 
with the unit of length. 

Statical Moments. 

Ml. The moment of a force, with reference to a point, is the 
measure of the effectiveness of the force in producing motion 
about the point. It is shown in treatises on Mechanics, that 
this is the product of the force and the perpendicular from tht 
point upon the line of application of the force. 



§ XVI.] STA TICAL MOMENTS. 211 

The moment of the sum of a number of forces about a 
given point is the sum of the moments of the forces. 

The statical moment of a body about a given point is the 
moment of its gravity ; the force of gravity being supposed to 
act upon every part of the body, and in parallel lines. 

178. In order to find the statical moment of a continuous 
body, we regard the body as generated geometrically in some 
convenient manner, and determine the corresponding differen- 
tial of the moment. 

In the case of a plane area, let the body be referred to 
rectangular axes, and let gravity be supposed to act in the 
direction of the axis of y. Then the abscissa of the point of 
application is the arm of the force when we consider the 
moment about the origin. Let us first suppose the area to be 
generated by the motion of the ordinate y. The differential of 
the area is then/ dx. The corresponding element of the sum, 

of which the integral y dx is the limiting value, see Art. 92, is 

J a 

y r Ax, (1) 

in which y,. is the ordinate corresponding to any value of x 
intermediate between a + (r — i)Ax, and a -f rAx. It is 
evident that the arm of the weight of the element (1) is such 
an intermediate value of x ; hence the moment of the ele- 
ment is 

x r yrAx (2) 

The whole moment is therefore the limiting value of a sum 
of the form 

2 x r y r Ax. 
In other words, it is the integral 

(3) 



J xydx, 



212 MECHANICAL APPLICATIONS. [Art. 1 78. 

in which the differential of the moment is the product of the 
differential of the area and the arm of the force, which in this 
case is the same for every point of the element. In other 
words, the moment of the differential is the differential of the 
moment. 

179. As an illustration, we find the moment of a semicircle 
(Fig. 36) about its centre. The area may be 
generated by the line 2jj/, moving from x = o to 
y|— ^ x = a. The equation of the circle being 

tf + / = ^, 

X 




Fig. 36. 



the differential of the area is 

2 Vid* — **) dx. 
The moment of this differential is 
2 V(a 2 — x*)x dx ; 
hence the whole moment is 

2 J" V(a* -x>)xdx = -- {a" - x*fY = — 



Centres of Gravity, 

180. If a force equal to the whole weight of a body be 
applied with an arm properly determined, its moment may be 
made equivalent to the whole statical moment of the body. 
If the force is in the direction of the axis of y, as in Fig. 36, we 
have, denoting this arm by x, 

x • Area = Moment, 

__ Moment 
x = Area " 



§ XVI] CENTRES OF GRAVITY. 21 3 

In like manner, supposing the force to act in the direction 
of the axis of x, we may determine y for the same body. 

It is shown in treatises on Mechanics that the point deter- 
mined by the two coordinates x and y, is independent of the 
position of the coordinate axis. This point is called the centre 
of gravity of the area. The centre of gravity of a volume is 
denned in like manner. 

181. The symmetry of the form of a body may determine 
one 01 more of the coordinates of its centre of gravity. Thus 
the centre of gravity of a circle or a sphere coincides with the 
geometrical centre, and the centre of gravity of a solid of revolu- 
tion is on the axis of revolution. The centre of gravity of the 
semicircle in Fig. 36, is on the axis of x ; hence to determine 
its position we have only to find ~x. Dividing the moment 
of the semicircle found in Art. 179 by the area %7ta 2 , we have 

_ 4a 

x = — . 

182. In finding the moment of the semicircle (Art. 179), we 
regarded the area as generated by the double ordinate 2y, and 
the differential of the moment was found by multiplying the 
differential of the area by x, which is the arm of the force for 
every point of the generating line. 

We may, however, derive the moment from the differential 
of area, 

*dy> (1) 

since the area may be generated by the motion of the abscissa 
x from y — — a to y = a. But in this case to find the moment 
of the differential we must multiply it by the distance of its 
centre of gravity from the given axis. The centre of gravity of 
the line x is evidently its middle point, hence the required arm 
is \x. Therefore the differential of the moment is 

x* dy . x 



214 MECHANICAL APPLICATIONS. [Art. 1 82. 

and consequently the whole moment is 

This result is identical with that derived in Art. 179. 

Polar Formula. 

(83. When polar formulae are employed, r and 6 being 
coordinates of the curved boundary of the area, the element is 
^r 2 dd. Since this element is ultimately a triangle, we employ 
the well known property of triangles ; that the centre of gravity 
is on a medial line at two-thirds the distance from the vertex 
to the base. 

The coordinates of the centre of gravity of the element are, 
therefore, 



2 2 

-rsmd and —rcosd. 



Hence we have the formula 

Ur cos 6 ^d6 2 [?* cos Odd 



Xz= 



•J 



t*d$ 3 [f*d6 



[^smddd 
and similarly J = 



3 \?*dd 



§ XVI.] POLAR FORMULAS. 215 

184. To illustrate, let us find the centre of gravity of the 
area enclosed by one loop of the lemniscata. 

r 2 = a 2 cos 20. 



[ 4 (cos20)*cos0af0 * 

Whence x = — - = 4? [ 4 (cos 20) 1 cos 6 dO. 

cos zOdB 

Put cos 20 = cos 2 $, whence sin (j> = 4/2 sin 6, 

and V2 cos 6 d6 = cos ^ ^, 



7T 

__2£2 fa 

~~T a h 



, . .. 2 V2 3 • I 7T V2 

cos 4 <^ ^ = a = — na. 

3 4-22 5 



Solids of Revolution. 

185. To find the centre of gravity of a solid of revolution, 
we take the axis of revolution as the axis of x y and the circle 
whose area is ny* as the generating element. Replacing y in 
equation (3), Art. 178, by this expression, we have for the stati- 
cal moment 

rb 

n xy 1 dx y 

J a 

and for the abscissa of the centre of gravity 

xy 1 dx 

x — ii 

b 

dx 



<> a 



2l6 MECHANICAL APPLICATIONS. [Art. 1 86. 

186. To illustrate, we find the centre of gravity of a spheri- 
cal segment whose height is h. In this case, taking the origin 
at the vertex of the segment, and denoting the radius of the 
sphere by <z, we have 

y 2 = 2ax — x*. 

[ h (2a**-3*)dx -ax* -IxtX r o 
Hence * = £ = 2 4_J^ ^h.Sa - 3 h 

J (2ox - x*)dx ax* - -X s ! 4 Z *~ k 

If the centre of gravity of the surface of the segment be re~ 
quired, since the differential of the surface is 2ny ds, we easily 
obtain the general formula 

xy ds 

eh * 

yds 

and, in this case the curve being a circle, y ds = adx; hence, 
substituting, we have 

x = ih. 



The Properties of Pappus. 

187. Let a solid be generated by the revolution of any plane 
figure about an exterior axis in its own plane. It is required 
to determine the volume and the surface thus generated. 

It is evident that this solid may also be generated by a 
variable circular ring whose centre moves along the axis of 
revolution ; denoting byj^i and y 2 corresponding ordinates of 



§ XVL] THE PROPERTIES OF PAPPUS. 2\*J 

the outer and inner circles respectively, the area of this ring is 
n (yi — y*)- Hence 

V= n\{y? - y{) dx = 2n\?±^{y 1 -y t ) dx. 



But this integral is the statical moment of the given figure, 

since j\ — y 2 is the generating element of its area, and —is 

the corresponding arm. Denoting the area of the figure by A, 
we may therefore write 

V= 2nyA ; 



that is, the volume is the product of the area of the figure and the 
path described by its centre of gravity. 

The surface (S) of this solid is, by Art. 149, 

S — 27T yds — 2ny \ds, 
if Jp denotes the ordinate of the centre of gravity of the arc s. 

Hence we have 5 = 2ny-arc ; 

that is, the surface is the product of the length of the arc into 
the path described by the centre of gravity. 

These theorems are frequently called the properties of Gul- 
dinus ; they are, however, due to Pappus, who published them 
1588. 

It is obvious that both theorems are true for any part of 
a revolution of the generating figure. 



2l8 MECHANICAL APPLICATIONS. [Ex. XVI. 



Examples XVI. 

i. Find the centre of gravity of the area enclosed between the 
parabola y* = \mx and the double ordinate corresponding to the 
abscissa a. 



2. Find the centre of gravity of the area between the semi-cubical 

parabola ay % = x 3 and the double ordinate which corresponds to the 

abscissa a. 

- 5<* 
x — ^~. 



3. Find the ordinate of the centre of gravity of the area between 
the axis of x and the sinusoid y = sin x, the limits being x = o and 
x=n. ~y—i7t. 

4. Find the coordinates of the centre of gravity of the area be* 
tween the axes and the parabola 



©H)'-- 



— a . - b 

x = — , and y — - 

5 5 



5. Find the centre of gravity of the area between the cissoid 

y 1 (a — x) — x 3 and its asymptote. 
Solution : — 

Denoting the statical moment by M and the area by A y 

M — 7 — — 2x 2 (a — xy* + 5 \ X* (a — xP dx 

Jo (a — x)~ Jo Jo 



A, hence 

6 o 



.*. M - ^ A, hence x— ^ 



§ XVI.] EXAMPLES. 219 

6. Find the centre of gravity of the area between the parabola 
r a = 4ax and the straight line y = mx. 

- 8a 1 - 2a 
x = — ;, and y = — . 

7. Find the centre of gravity of the segment of an ellipse cut off 
by a quadrantal chord. 

— 2a . - 2 b 
x = - . and y = - • . 

3 71 — 2* y 3 7T-2 

8. Given the cycloid, 

y — a ( 1 — cos tp), x — a (ip — sin ip) , 

find the distance of the centre of gravity of the area from the base. 

r. _5* 



9. Find the centre of gravity of the area enclosed between the 
positive directions of the coordinate axes and the four-cusped hypo- 
cycloid 



Put x = a cos 3 6, andy = a sin 3 0. 



- - 2?6tf 

x= y = — 

315* 



10. Find the centre of gravity of the area enclosed by the cardioid 
r =a(i — cos 0). 

S a 

n. Find the centre of gravity of the sector of a circle whose radius 

is a, the angle of the sector being 2 a. 

— 2 a sin ol 
Use the method of Art. 183. x = 



220 MECHANICAL APPLICATIONS. [Ex. XVI. 

12. Find the centre of gravity of the segment of a circle, the angle 
subtended being 2a and the radius of the circle 0. 

Solution : — 

2 (0 s — x*)xdx _, 



J a cos a 



20 s sin 3 a Chord 3 
x = 



Area 3 Area 12 Area 

13. Find the centre of gravity of a circular ring, the radii being a 
and «i, and the angle subtended 2<z. 

- _ 2 a % — a? sin or 

3 0— 01 « 

14. Find the centre of gravity of a circular arc, whose length is 2s. 

Solution : — 

We have in this case, taking the origin at the centre and the axis 
of x bisecting the arc, 



L- 



ds 



1> 



Put x — a cos B, then ds = a de, and denoting by a the 

angle subtended by s, we have 



,r 



cos 6 0i9 

sin <* __ c 

9 



2s a a 

2c being the chord. 



§XVL] EXAMPLES. 221 

15. Find the coordinates of the centre of gravity of arc of the semi- 
cycloid whose equations, referred to the vertex, are 

x = a(i— costp), and y = a(if> + sin tp). 



x = — , and y 



= (-!)- 



16. Find the centre of gravity of the arc between two successive 
cusps of the four-cusped hypocycloid 



a , 2. 2. 



_ _ 2a 

X=y = - 



17. Find the position of the centre of gravity of the arc of the semi- 
cardioid 

r = a (1 — cos 0). 

40-40 

x = , and y — — . 

5 5 

18. A serai-ellipsoid is formed by the revolution of a semi-ellipse 
about its major axis ; find the distance of the centre of gravity of the 
solid from the centre of the ellipse. 

r_ 3* 



19. Find the centre of gravity of a frustum of a paraboloid of 
revolution having a single base, h denoting the height of the frustum. 

- zh 

x = — . 
3 

20. A paraboloid and a cone have a common base and vertices at 
the same point ; find the centre of gravity of the solid enclosed 
between them. 

The centre of gravity is the middle point of the axis. 



MECHANICAL APPLICATIONS. [ Ex - XVI - 

21. Find the centre of gravity of a hyperboloid whose height is h» 
the generating curve being 

y 1 — m {2ax + x 2 ). 

- _ h &a + 3/1 

4 30 + h 

22. Find the centre of gravity of the solid formed by the revolution 
of the sector of a circle about one of its extreme radii. 

The height of the cone being denoted by h y and the radius of the 
circle by a, we have 

* = |(« +h). 

23. Find the centre of gravity of the solid formed by the revolution 
about the axis of x of the curve 

a?y = ax 2 — x* t 

between the limits o and a. 

24. A solid is formed by revolving about its axis the cardioid 

r — a (1 — cose) ; 
find the distance of the cusp from the centre of gravity. 

— Ad 

X z= — • 

5 

25. Determine the position of the centre of gravity of the volume 
included between the surfaces generated by revolving about the axis 
of x the two parabolas 

y = mx, and y* = m' (a — x). 

— a m + 2m 

# = - : r- 

3 m + m 



§ XVI.] EXAMPLES. 223 

26. Find the centre of gravity of a rifle bullet consisting of a cylin- 
der two calibers in length, and a paraboloid one and a half calibers in 
length having a common base, the opposite end of the cylinder con- 
taining a conical cavity one caliber in depth with a base equal in size 
to that of the cylinder. 

The distance of the centre of gravity from the base of 
the bullet is if! calibers. 

27. A solid formed by the revolution of a circular segment about 
its chord is cut in halves by a plane perpendicular to the chord ; 
determine the centre of gravity of one of the halves. This solid is 
called an ogival. 

Denoting by 2a the angle subtended by the chord, and by a the 
radius of the circle, the distance of the centre of gravity from the 
base is 

- _ a 44 sin 2 a + sin 2 2^+32 (cos 2 a — cos a) 
1 6 sin a (2 4- cos" a) — $a cos a 

28. Find the centre of gravity of the surface of the paraboloid 
formed by the revolution about the axis of x of the parabola 

/ - 4mx, 
a denoting the height of the paraboloid. 

- _ I (30 "~ 2m ) ( a + w)f + 2M S 

x — — . — . 

5 (a 4- m) z — m* 

29. Find the centre of gravity of the surface generated by the revo- 
lution of a semi-cycloid about its axis, the equations of the curve 
being 

x = a (1 — cos ip), and y = a (ip + sin tp). 

- __ 2a I 5 7r ~~ 8 
*~ *5' 3^-4 ' 



22 4 MECHANICAL APPLICATIONS. L Ex - XVI. 

30. Find the centre of gravity of the surface generated by the revo- 
lution about its axis of one of the loops of the lemniscata 



— 2+V2 
x — r— a- 



31. A cardioid revolves about its axis ; find the centre of gravity 
of the surface generated, the equation of the cardioid being 



r = a (1 — cos 9). 



5°a 
63' 



32. A ring is generated by the revolution of a circle about an axis 
in its own plane ; c being the distance of the centre of the circle 
from the axis, and a the radius, determine the volume and surface 
generated. 

V—2 7t i cd\ and S — ^n^ca. 

33. A triangle revolves about an axis in its plane ; a lt a 2 , and a a> 
denoting the distances of its vertices from the axis, determine the vol- 
ume generated. 

27TA . v 

V = (#i + 2 + a 3 ). 

3 

34. Find the volume of a frustum of a cone, the radii of the bases 
being a x and a 2 , and the height h, 

V= — (ax + a x a<i. + aT). 
3 



35. Find the volume and surface generated by the revolution of a 
cycloid about its base. 

ir 23 jo 647T0 2 

V= $7t*a\ and S = — - — -. 



§ XVII.] MOMENTS OF INERTIA. 225 

XVII. 

Moments of Inertia. 

188. When a body rotates about a fixed axis, the velocit) 
of a particle at a distance r from the axis is 

doo 

in which 00 is the angle of rotation. The force which acting 
for a unit of time would produce this motion in a mass m is 
measured by the momentum 

doo 

mr —r . 
dt 

The moment of this force about the axis is therefore 

o doo 

mr* —r . 
dt 

The sum of these moments for all the parts of a rigid system is 

at 

since the angular velocity, — , is constant. In the case of a 

a z 



doo 
~dt 



continuous body this expression becomes 

r 2 dtn, 

in which dm is the differential of the mass. The factor 

Wdfn, 



226 MECHANICAL APPLICATIONS. [Art. 1 88* 

which depends upon the shape of the body, is called its mo- 
ment of inertia , and is denoted by /. 

189. When the body is homogeneous, dm is to be taken 
equal to the differential of the line, area, or volume, as the case 
may be. For example, in finding the moment of inertia of a 
straight line whose length is 2a, about an axis bisecting it at 
right angles, we let x denote the distance of any point from 
the axis ; then dm = dx, hence we have 

J-a 3 I2 

Again, in finding the moment of inertia of the semi-circle in 
figure 36, about the axis of y, let dm — 2ydx\ then, since every 
point of the generating line is at the distance x from the axis, 
the moment of inertia is 

I = 2\ yx 2 dx = 2 V(a 2 — x 2 ) x 2 dx . 

J c Jo 

Putting x = asm 6, we have 

7T 

= 2a" \ 



I=2a 4 V cos 2 6 sin 2 6 d6 = ^--. 



The Radius of Gyration. 

190. If the whole mass of the body were situated at the 
distance k from the axis, its moment of inertia would be &tn. 
Now, if k is so determined that this moment shall be equal to 
the actual moment of inertia of the body, the value of k is the 
radius of gyration of the body with reference to the given 
axis. Hence 

£2 __ Moment of inertia 
Mass 



§ XVII.] 7WE RADIUS OF GYRATION. 22J 

Thus, for the radius of gyration of the line 2#, whose moment 
of inertia is found in the preceding article, we have 



& = - , or k — — ; 

3 V3 

and for the radius of gyration of the semi-circle, whose area 
is ^7ta 2 , 

1? — - , or k— -. 

4 2 

It is evident that this expression is also the radius of gyra- 
tion of the whole circle about a diameter, for the moment of 
inertia of the circle is evidently double that of the semi-circle, 
and its area is also double that of the semi-circle. 

191. It is sometimes convenient to use modes of generating 
the area or volume, other than those involving rectangular 
coordinates. For example, let it be required to find the radius 
of gyration of a circle whose radius is a, about an axis passing 
through its centre and perpendicular to its plane. This circle 
may be generated by the circumference of a variable circle 
whose radius is r, while r passes from o to a. The differential 
of the area is then 2nr dr y and the moment is 



I — 27t\ r 3 dr = — , 



Dividing by the area of the circle, we have 



B = a -. 



192. Again, to find the radius of gyration of a sphere 
whose radius is a about a diameter. In order that all points 
of the elements shall be at the same distance from the axis 



228 MECHANICAL APPLICATIONS. [Art. I92. 

we regard the sphere as generated by the surface of a cylinder 
whose radius is x, and whose altitude is 2y. The surface of 
this cylinder is therefore \nxy. The differential of the volume 
is ^nxy dx, and the moment of inertia is 



/ = 47T J x z y dx = 47c V(a* —x 2 ) X s dx. 
Putting x = a sin 6, 

IT 

I = 47ta 5 f 2 sin 3 cos 2 dd = — . 



Dividing by - — , the volume of the sphere, we have 



^ = 2 ^ 



Radii of Gyration about Parallel Axes 

193. The moment of inertia of a body about any axis exceeds 
its moment of inertia about a parallel axis passing through the 
centre of gravity, by the product of the mass and the square of 
the distance between the axes. 

Let h be the distance between the axes. Pass a plane 
through the element dm perpendicular to the axes, and let r 
and r x be the distances of the element from the axes. Then, 
r, r lf and h form a triangle ; let 6 be the angle at the axis 
passing through the centre of gravity, then 

r>= ri + B — 2r x h cos 6. . , * , B (1) 



§ XVII. j RADII OF GYRATION ABOUT PARALLEL AXES. 229 

The moment of inertia is therefore 

I r^dm — rl dm + ffm — 2h\r x cos 6 dm . . . (2) 

Now r x and 6 are the polar coordinates of dm, in the plane 
which is passed through the element; hence the last integral in 
equation (2) is equivalent to 



-2k\ 



dm. 



But x dm is the statical moment of the body about the axis 

passing through the centre of gravity. Now from the defini- 
tion of the centre of gravity, this moment is zero ; hence, 
equation (2) reduces to 

r 2 dm = rf dm + Iftm (3 

Introducing the radii of gyration, we have also 

# = # + # (4) 

194. As an application of this result, we shall now find the 
moment of inertia of a cone whose height is h, and the radius 
of whose base is a, about an axis passing through its vertex 
perpendicular to its geometrical axis. Taking the origin at 
the vertex of the cone, the axis of x coincident with the geo- 
metrical axis, and a circle perpendicular to this axis as the 
generating element, we have for the area of this element ny 1 , 
and for its radius of gyration about a diameter parallel to 

y 

the given axis, - (Art. 190). 



230 MECHANICAL APPLICATIONS. [Art. 194. 

The distance between these axes being x, the proposition 

proved in the preceding article gives an expression for the 

radius of gyration of the element about the given axis ; viz., 

y 2 
x* + - . Replacing t- 2 , in the general expression for / (Art. 

4 
188), by this expression, and substituting for dm the differen- 
tial ny* dx y we have 





I 


= n\(x> + } ^ydx i 






in which y — 


ax 

: T' 


Therefore 






/ = 


7ta*[ h ( 

JFlV 


+ 4*7 5 


(■ 


a 2 \ 


and since 




j Z _ 7ta % k 
3 ' 







^U?***)' 



To find the square of the radius of gyration about a 
parallel axis through the centre of gravity, we have 



4 2 = -^-U 2 

20 



=U' 



■)-.©' 



To find the moment of inertia of a right cone about its 

geometrical axis we employ the same generating element as 

before ; but in this case the square of the radius of gyration is 

r 2 

- . Hence 

2 



* =IV * = %]/**■> 



§ XVI 1.] RADII OF GYRA TION ABOUT PARAILEL AXES. 23 I 



therefore 



/= , whence &=~— 

10 10 



Polar Moments of Inertia. 



195. In the case of a plane area, when the axis of rotation 
passes through the origin, we have 

r 2 = x 2 + y, . hence r 2 dm — (x 2 + y 2 ) dm, 

therefore /= Ix 2 dm + M/ 2 dm ; 

that is, the sum of the moments of inertia of a plci7ie area about 
two axes in its own plane at right angles to each other is equal to 
the moment of inertia about an axis through the origin perpendicu- 
lar to the plane. I in the above equation is called the polar 
moment of inertia. 

In the case of the circle, since the moment is the same 
about every diameter, the polar moment is twice the moment 
about a diameter ; that is, denoting the former by Ip and the 
latter by I a , we have 



See Art. 191. 



T na* 



Examples XVII. 



1. Find the radius of gyration of a circular arc (2s) about a radius 
passing through its vertex. 



232 MECHANICAL APPLICATIONS. [Ex. XVI L 

Solution : — 

Taking the origin at the centre, and the axis of x bisecting the arc, 
and denoting by 2 a the angle subtended by 2s, we have 

mk a = [* / ds — a % | sin 2 6 do. 

72 a* ( sin 2<x\ 
m = 2aot . . >£ = — ( 1 

2 \ 2a J 



2. Find the radius of gyration of the same arc about the axis of y, 
and thence about a perpendicular axis through the centre of the 
circle. k = a. 

3. Find the radius of gyration of the same arc about an axis through 
its vertex perpendicular to the plane of the circle. 

See Ex. XVI., 14, and denote by c the subtending chord. 

o / c 

k = 2a 

2S 

4. Find the moment of inertia of the chord of a circular arc, in 
terms of the diameter parallel to it, and its angular distance from this 
diameter. 

See Arts. 189 and 193. / = — (3 cos a — cos 3a) . 

24 

5. Find the radius of gyration of an ellipse about an axis through 
its centre perpendicular to its plane. 

Find the radius of gyration about the major axis and about the minor 
axis, and apply Art. 195. 

tf = ±(a' 2 4- ?). 

6. Find the radius of gyration of an isosceles triangle about a per- 
pendicular let fall from its vertex upon the base (2b). 

e = t 

6" 



§ XVI I.] EXAMPLES. 233 

7. Find the radius of gyration about the axis of the curve, of the 
area enclosed by the two loops of the lemniscata 



8. Find the radius of gyration of a right triangle, whose sides are a 
and b, about an axis through its centre of gravity perpendicular to its 
plan- 

a- + tf 



k* = 



18 



9. Find the radius of gyration of a portion of a parabola bounded 
by a double ordinate perpendicular to the axis, about a perpendicular 
to its plane passing through its vertex. 

? = 4 «* + */. 

10. Find the radius of gyration of a cylinder about a perpendicular 
that bisects its geometrical axis, 2/ being the length of the cylinder, 
and a the radius of its base. 

a" I* 

— + - . 
4 3 



? = - + 



11. Find the radius of gyration of a concentric spherical shell about 
a tangent to the external sphere, the radii being a and b. 

p- la* -$(£$ -2b h 

12. Find the radius of gyration of a paraboloid of revolution about 
its axis, in terms of the radius (b) of the base. 

3 

13. Find the moment of inertia of an ellipsoid about one of its 
principal axes. 

J= Vtabc_ 

'5 



234 MECHANICAL APPLICATIONS. [Ex. XVII. 

14. Find the radius of gyration of a symmetrical double convex lens 
about its axis, a being the radius of the circular intersection of tne 
two surfaces, and b the semi-axis. 

* ~ io(^ 2 + 3 * 2 ) ' 

15. Find the radius of gyration of the same lens about a diameter 
to the circle in which the spherical surfaces intersect. 

, a _ 10a 4 + 15^ + 7/ * 

* "" 20(£ 2 + 3 a ) ~ ' 



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BOTANY. 

Gardening for Ladies, Etc. 

Baldwin's Orchids of New England 8vo, 1 50 

Loudon's Gardening for Ladies. (Downing.) 12mo, 1 50 

Thome's Structural Botany . . 18mo, 2 25 

We^termaier's General Botany. (Schneider.) 8vo, 2 00 

BRIDGES, ROOFS, Etc. 

Cantilever — Draw — Highway — Suspension. 
(See also Engineering, p. 6.) 

Boiler's Highway Bridges 8vo, 2 00 

* " The Thames River Bridge 4to, paper, 5 00 

Burr's Stresses in Bridges. 8vo, 3 50 

Crehore's Mechanics of the Girder 8vo, 5 00 

Dredge's Thames Bridges 7 parts, per part, 1 25 

Du Bois's Stresses iu Framed Structures 4to, 10 00 

Foster's Wooden Trestle Bridges 4to, 5 00 

Greene's Arches in Wood, etc 8vo, 2 50 

" Bridge Trusses 8vo, 2 50 

" Roof Trusses 8vo, 1 25 

Howe's Treatise on Arches 8vo, 4 00 

Johnson's Modern Framed Structures .4to, 10 00 

Merriman & Jacoby's Text-book of Roofs and Bridges. 

Part I. , Stresses 8vo, 2 50 

Merriman & Jacoby's Text-book of Roofs and Bridges. 

Part II.. Graphic Statics 8vo, 2 50 

Merriman & Jacoby's Text-book of Roofs and Bridges. 

Part III., Bridge Design Svo, 2 50 

Merriman & Jacoby's Text- book of Roofs and Bridges. 

Part IV., Continuous, Draw, Cantilever, Suspension, and 

Arched Bridges 8vo, 2 50 

* Morison's The Memphis Bridge Oblong 4to, 10 00 

4 



Waddell's Iron Highway Bridges 8vo, $4 00 

" De Pontibus (a Pocket-book for Bridge Engineers). 

Wood's Construction of Bridges and Roofs 8vo, 2 00 

Wright's Designing of Draw Spans 8vo, 2 50 

CHEMISTRY. 

Qualitative — Quantitative- Organic — Inorganic, Etc. 

Adriance's Laboratory Calculations 12rno, 

Allen's Tables for Iron Analysis 8vo, 

Austen's Notes for Chemical Students 12mo, 

Bolton's Student's Guide in Quantitative Analysis 8vo, 

Classen's Analysis by Electrolysis. (Herrick and Boltwood.).8vo, 

Crafts's Qualitative Analysis. (Schaeffer. ) 12mo, 

Drechsel's Chemical Reactions. (Merrill.) 12mo, 

Eresenius's Quantitative Chemical Analysis. (Allen.) 8vo, 

" Qualitative " V (Johnson.) 8vo, 

(Wells) Trans. 16th. 

German Edition 8vo, 

Euerte's Water and Public Health 12mo, 

Gill's Gas and Fuel Analysis 12mo, 

Hammarsten's Physiological Chemistry. (Maudel.) 8vo, 

Helm's Principles of Mathematical Chemistry. (Morgan). 1 2 ino, 

Kolbe's Inorganic Chemistry „ 12mo, 

Ladd's Quantitative Chemical Analysis 12mo, 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 

Mandel's Bio-chemical Laboratory 12mo, 

Mason's Water-supply 8vo, 

" Analysis of Potable Water. {In the press.) 

Miller's Chemical Physics 8vo, 

Mixter's Elementary Text-book of Chemistry , 12mo, 

Morgan's The Theory of Solutions and its Results 12mo, 

Nichols's Water-supply (Chemical and Sanitary) 8vo, 

O'Brine's Laboratory Guide to Chemical Analysis 8vo, 

Perkins's Qualitative Analysis ,12mo, 

Pinner's Organic Chemistry. (Austen.) 12mo, 

Poole's Calorific Power of Fuels 8vo, 

Ricketts and Russell's Notes on Inorganic Chemistry (Non- 

metallic) , Oblong 8vo, morocco, 75 



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Ruddimaii's Incompatibilities in Prescriptions 8vo, $2 00 

Sehimpf s Volumetric Analysis 12mo, 2 50 

Spencer's Sugar Manufacturer's Handbook . 12mo, morocco flaps, 2 00 

" Handbook for Chemists of Beet Sugar House. 

12mo, morocco, 3 00 

Stockbridge's Rocks and Soils 8vo, 2 50 

Troilius's Chemistry of Iron 8vo, 2 00 

Wells's Inorganic Qualitative Analysis 12mo, 1 50 

" Laboratory Guide in Qualitative Chemical Analysis, 8vo, 1 50 

"Wiechmann's Chemical Lecture Notes 12mo, 3 00 

" Sugar Analysis ., . . 8vo, 2 50 

Wulling's Inorganic Phar. and Med. Chemistry 12mo, 2 00 

DRAWING. 

Elementary — Geometrical — Topographical. 

Hill's Shades and Shadows and Perspective 8vo, 

MacCord's Descriptive Geometry 8vo, 

' ' Kinematics 8 vo, 

" Mechanical Drawing 8vo, 

Mahan's Industrial Drawing. (Thompson.) 2 vols., 8vo, 

Reed's Topographical Drawing. (II. A.) 4to, 

Reid's A Course in Mechanical Drawing 8vo. 

" Mechanical Drawing and Elementary Machine Design. 

8vo. 

Smith's Topographical Drawing. (Macmillan.) 8vo, 

Warren's Descriptive Geometry 2 vols., 8vo, 

" Drafting Instruments 12mo, 

Free-hand Drawing .... 1 2mo, 

" Higher Linear Perspective 8vo, 

" Linear Perspective 12mo, 

" Machine Construction * 2 vols., 8vo, 

" Plane Problems , 12mo, 

" Primary Geometry 12mo, 

" Problems and Theorems 8vo, 

" Projection Drawing 12mo, 

" Shades and Shadows 8vo, 

" Stereotomy— Stone Cutting 8vo, 

Whelpley's Letter Engraving 12mo, 

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ELECTRICITY AND MAGNETISM. 

Illumination— Batteries— Physics. 

Anthony and Brackett's Text-book of Physics (Magie). . . .8vo, $4 00 

Barker's Deep-sea Soundings 8vo, 2 00 

Benjamin's Voltaic Cell 8vo, 3 00 

History of Electricity 8vo 3 00 

Cosmic Law of Thermal Repulsion 18mo, 75 

Crehore and Squier's Experiments with a New Polarizing Photo- 
Chronograph 8vo, 3 00 

* Dredge's Electric Illuminations. . . .2 vols., 4to, half morocco, 25 00 

Vol. II 4to, 7 50 

Gilbert's De magnete. (Mottelay.) 8vo, 2 50 

Holman's Precision of Measurements 8vo, 2 00 

Michie's Wave Motion Relating to Sound and Light, 8vo, 4 00 

Morgan's The Theory of Solutions and its Results 12mo, 1 00 

Niaudet's Electric Batteries. (Fishback.) 12mo, 2 50 

Reagan's Steam and Electrical Locomotives 12mo, 2 00 

Thurston's Stationary Steam Engines for Electric Lighting Pur- 
poses 12mo, 1 50 

Tillman's Heat 8vo, 1 50 

ENGINEERING. 

Civil — Mechanical — Sanitary, Etc. 

{See also Bridges, p. 4 ; Hydraulics, p. 8 ; Materials of En- 
gineering, p. 9 ; Mechanics and Machinery, p. 11 ; Steam Engines 
and Boilers, p. 14.) 

Baker's Masonry Construction .* . 8vo, 5 00 

" Surveying Instruments 12mo, 3 00 

Black's U. S. Public Works 4to, 5 00 

Brook's Street Railway Location .. 12mo, morocco, 1 50 

Butts's Engineer's Field-book - 12mo, morocco, 2 50 

Byrne's Highway Construction 8vo, 7 50 

11 Inspection of Materials and Workmanship. 12mo, mor. 

Carpenter's Experimental Engineering 8vo, 6 00 

Church's Mechanics of Engineering — Solids and Fluids 8vo, 6 00 

Notes and Examples in Mechanics 8vo, 2 00 

Crandall's Earthwork Tables 8vo, 1 50 

' ' The Transition Curve 12mo, morocco, 1 50 

7 



* Dredge's Perm. Railroad Construction, etc. . . Folio, half mor., $20 00 

* Drinker's Tunnelling 4to, half morocco, 

Eissler's Explosives — Nitroglycerine and Dynamite 8vo, 

Fowler's Coffer-dam Process for Piers . . 8vo.. 

Gerhard's Sanitary House Inspection 16mo, 

Godwin's Railroad Engineer's Field-book. 12mo, pocket-bk. form, 

Gore's Elements of Geodesy . « 8vo, 

Howard's Transition Curve Field-book., . . .12mo, morocco flap, 

Howe's Retaining Walls (New Edition.) e . . . .12mo, 

Hudson's Excavation Tables. Vol. II , 8vo, 

Hutton's Mechanical Engineering of Power Plants 8vo, 

Johnson's Materials of Construction 8vo, 

" Stadia Reduction Diagram. .Sheet, 22£ X 28£ inches, 

" Theory and Practice of Surveying 8vo, 

Kent's Mechanical Engineer's Pocket-book 12mo, morocco, 

Kiersted's Sewage Disposal 12mo, 

Kirkwood's Lead Pipe for Service Pipe 8vo, 

Mahan's Civil Engineering. (Wood.) 8vo, 

Merriman and Brook's Handbook for Surveyors. . . .12mo, mor., 

Merriman's Geodetic Surveying 8vo, 

" Retaining Walls and Masonry Dams 8vo, 

Mosely's Mechanical Engineering. (Mahan.) 8vo, 

Nagle's Manual for Railroad Engineers 12mo, morocco, 

Pattou's Civil Engineering ,8vo, 

" Foundations 8vo, 

Rockwell's Roads and Pavements in France 12mo, 

Ruffuer's Noa-tidal Rivers. 8vo, 

Searles's Field Engineering 12mo, morocco flaps, 

" Railroad Spiral , 12mo, morocco flaps, 

Siebert and Biggin's Modern Stone Cutting and Masonry. . .8vo, 

Smith's Cable Tramways. . .„ 4to, 

" Wire Manufacture and Uses , 4to, 

Spalding's Roads and Pavements 12mo, 

" Hydraulic Cement. . . , 12mo, 

Thurston's Materials of Construction 8vo, 

* Trautwiue's Civil Engineer's Pocket-book. ..12mo, mor. flaps, 

* ' ' Cross-section Sheet, 

* " Excavations and Embankments 8vo, 

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* Trautwiue's Laying Out Curves 12nio, morocco, $2 50 

Waddell's De Pontibus (A Pocket-book for Bridge Engineers). 

12mo, morocco, 3 00 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 50 

" Law of Field Operation in Engineering, etc 8vo. 

Warren's Stereotomy— Stone Cutting 8vo, 2 50 

Webb s Engineering Instruments 12mo, morocco, 1 00 

Wegmann's Construction of Masonry Dams 4to, 5 00 

Wellington's Location of Railways 8vo, 5 00 

Wheeler's Civil Engineering 8vo, 4 00 

Wolff's Windmill as a Prime Mover 8vo, 3 00 

HYDRAULICS. 

Water-wheels— Windmills— Service Pipe— Drainage, Etc. 
{See also Engineering, p. 6.) 
Bazin's Experiments upon the Contraction of the Liquid Vein 

(Trautwiue) 8vo, 

Bovey's Treatise on Hydraulics. . . , 8vo, 

Coffin's Graphical Solution of Hydraulic Problems 12mo, 

Ferrel's Treatise on the Winds, Cyclones, and Tornadoes. . .8vo, 

Fuerte's Water and Public Health 12mo, 

Ganguillet&Kutter'sFlow of Water. (Hering&Trautwine.).8vo, 

Hazen's Filtration of Public Water Supply 8vo, 

Herschel's 115 Experiments 8vo, 

Kiersted's Sewage Disposal 12mo, 

Kirkwood's Lead Pipe for Service Pipe . „ 8vo, 

Mason's Water Supply 8vo, 

Merriman's Treatise on Hydraulics. . , 8vo, 

Nichols's Water Supply (Chemical and Sanitary) 8vo, 

Ruffner's Improvement for Non-tidal Rivers 8vo, 

Wegmann's Water Supply of the City of New York 4to, 

Weisbach's Hydraulics. (Du Bois.) 8vd, 

Wilson's Irrigation Engineering 8vo, 

Hydraulic and Placer Mining 12mo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Theory of Turbines 8vo, 

MANUFACTURES. 

Aniline — Boilers— Explosives— Iron— Sugar — Watches — 
Woollens, Etc. 

Allen 's Tables for Iron Analysis 8vo, 3 00 

Beaumont's Woollen and Worsted Manufacture 12mo, 1 50 

Bollaud's Encyclopaedia of Founding Terms 12mo, 3 00 

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Bolland's The Iron Founder 12mo, 

" " " Supplement..., 12rno, 

Booth's Clock and Watch Maker's Manual 12mo, 

Bouvier's Handbook on Oil Painting 12mo, 

Eissler's Explosives, Nitroglycerine and Dynamite. . 8vo, 

Ford's Boiler Making for Boiler Makers 18mo, 

Metcalfe's Cost of Manufactures 8vo, 

Metcalf 's Steel— A Manual for Steel Users 12nio, 

Reimann's Aniline Colors. (Crookes.) 8vo, 

*Reisig's Guide to Piece Dyeing 8vo, 

Spencer's Sugar Manufacturer's Handbook. . . .12mo, mor. flap, 
" Handbook for Chemists of Beet Houses, 

12mo, mor. flap, 

Svedelius's Handbook for Charcoal Burners 12mo, 

The Lathe and Its Uses 8vo, 

Thurston's Manual of Steam Boilers 8vo, 

Walke's Lectures on Explosives . 8vo, 

West's American Foundry Practice 12mo, 

" Moulder's Text-book 12mo, 

Wiechmann's Sugar Aualysis 8vo, 

Woodbury's Fire Protection of Mills 8vo, 



MATERIALS OF ENGINEERING. 

Strength — Elasticity — Resistance, Etc. 
{See also Engineering, p. 6.) 

Baker's Masonry Construction 8vo, 

Beardslee and Kent's Strength of Wrought Iron 8vo, 

Bovey's Strength of Materials 8vo, 

Burr's Elasticity and Resistance of Materials 8vo, 

Byrne's Highway Construction , 8 vo, 

Carpenter's Testing Machines and Methods of Testing Materials. 

Church's Mechanics of Engineering — Solids and Fluids 8vo, 

Du Bois's Stresses in Framed Structures 4to, 

Hatfield's Transverse Strains 8vo, 

Johnson's Materials of Construction 8vo, 

Lanza's Applied Mechanics 8vo, 

Merrill's Stones for Building and Decoration 8vo, 

Merriman's Mechanics of Materials : . . . . 8vo, 

" Strength of Materials 12mo, 

Patton's Treatise on Foundations 8vo, 

Rockwell's Roads and Pavements in France 12mo, 

Spalding's Roads and Pavements ... .12mo, 

Thurston's Materials of Construction , 8vo, 

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Thurston's Materials of Engineering 3 vols., 8vo, $8 00 

Vol. I, Non-metallic 8vo, 2 00 

Vol. II., Iron and Steel 8vo, 3 50 

Vol. III., Alloys, Brasses, and Bronzes 8vo, 2 50 

Weyrauch's Strength of Iron and Steel. (Du Bois.) 8vo, 150 

Wood's Resistance of Materials 8vo, 2 00 

MATHEMATICS. 

Calculus— Geometry— Trigonometry, Etc. 

Baker's Elliptic Functions 8vo, 1 50 

Ballard's Pyramid Problem 8vo, 1 50 

Barnard'3 Pyramid Problem 8vo, 1 50 

Bass's Differential Calculus , 12mo, 4 00 

Brigg's Plane Analytical Geometry 12mo, 1 00 

Chapman's Theory of Equations 12mo, 1 50 

Chessin's Elements of the Theory of Functions. 

Compton's Logarithmic Computations 12mo, 1 50 

Craig's Linear Differential Equations 8vo, 5 00 

Davis's Introduction to the Logic of Algebra 8vo, 1 50 

Halsted's Elements of Geometry ...8vo, 1 75 

" Synthetic Geometry 8vo, 1 50 

Johnson's Curve Tracing 12mo, 1 00 

" Differential Equations— Ordinary and Partial 8vo, 3 50 

" Integral Calculus 12mo, 1 50 

" " " Unabridged. 

" Least Squares 12mo, 1 50 

Ludlow's Logarithmic and Other Tables. (Bass.) 8vo, 2 00 

Trigonometry with Tables. (Bass.) .8vo, 3 00 

Mahan's Descriptive Geometry (Stone Cutting) 8vo, 1 50 

Merriman and Woodward's Higher Mathematics 8vo, 5 00 

Merriman's Method of Least Squares 8vo, 2 00 

Parker's Quadrature of the Circle 8vo, 2 50 

Rice and Johnson's Differential and Integral Calculus, 

2 vols, in 1, 12mo, 2 50 

Differential Calculus 8vo, 3 00 

" Abridgment of Differential Calculus.... 8vo, 150 

Searles's Elements of Geometry 8vo, 1 50 

Totten's Metrology 8vo, 2 50 

Warren's Descriptive Geometry 2 vols., 8vo, 3 50 

' ' Drafting Instruments 12mo, 1 25 

" Free-hand Drawing 12mo, 1 00 

" Higher Linear Perspective 8vo, 3 50 

" Linear Perspective 12mo, 1 00 

11 Primary Geometry 12mo, 75 

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Warren's Plane Problems, 12mo, $1 25 

" Problems and Theorems 8vo, 2 50 

" Projection Drawing 12mo, 150 

Wood's Co-ordinate Geometry 8vo, 2 00 

" Trigonometry 12mo, 1 00 

Woolf s Descriptive Geometry Royal 8vo, 3 00 

MECHANICS-MACHINERY. 

Text-books and Practical Works. 
(See also Engineering, p. 6.) 

Baldwin's Steam Heating for Buildings 12mo, 

Benjamin's Wrinkles and Recipes 12mo, 

Carpenter's Testing Machines and Methods of Testing 

Materials » 8vo. 

Chordal's Letters to Mechanics 12mo, 

Church's Mechanics of Engineering, 8vo, 

" Notes and Examples in Mechanics . . .• 8vo, 

Crehore's Mechanics of the Girder 8vo, 

Cromwell's Belts and Pulleys 12mo, 

" Toothed Gearing .12mo, 

Compton's First Lessons in Metal Working 12mo, 

Dana's Elementary Mechanics 12mo, 

Dingey's Machinery Pattern Making 12mo, 

Dredge's Trans. Exhibits Building, World Exposition, 

4to, half morocco, 

Du Bois's Mechanics. Vol. I., Kinematics , 8vo, 

Vol. II.. Statics 8vo, 

Vol III., Kinetics, 8vo, 

Fitzgerald's Boston Machinist , 18mo, 

Flather's Dynamometers 12mo, 

" Rope Driving 12mo, 

Hall's Car Lubrication 1 2mo, 

Holly's Saw Filing 18mo, 

Johnson's Theoretical Mechanics. An Elementary Treatise. 
(In the press.) 

Jones Machine Design. Part I., Kinematics 8vo, 1 50 

" " " Part II., Strength and Proportion of 

Machine Parts. 

Lanza's Applied Mechanics 8vo, 7 50 

MacCord's Kinematics 8vo, 5 00 

Merriman's Mechanics of Materials 8vo, 4 00 

Metcalfe's Cost of Manufactures 8vo, 5 00 

Michie's Analytical Mechanics Svo, 4 00 

Mosely's Mechanical Engineering. (Mahan.) 8vo. 5 00 

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Richards's Compressed Air 12mo, $1 50 

Robinson's Principles of Median ism 8vo, 3 00 

Smith's Press-working of Metals 8vo, 5» 00 

Tbe Latbe and Its Uses 8vo, 6 00 

Thurston's Friction and Lost Work 8vo, 3 00 

Tbe Animal as a Machine , 12mo, 1 00 

Warren's Machine Construction 2 vols., 8vo, 7 50 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.)..8vo, 5 00 
" Mechanics of Engineering. Vol III., Part I., 

Sec. I. (Klein.) 8vo, 5 00 

Weisbach's Mechanics of Engineering Vol. III., Part I., 

Sec. II (Klein.) 8vo, 5 00 

Weisbach's Steam Engines. (Du Bois.) 8vo, 5 00 

Wood's Analytical Mechanics 8vo, 3 00 

" Elementary Mechanics 12mo, 125 

" " " Supplement and Key 1 25 



METALLURGY. 

Iron— Gold— Silver — Alloys, Etc. 

Allen's Tables for Iron Analysis 8vo, 

Egleston's Gold and Mercury 8vo, 

" Metallurgy of Silver 8vo, 

* Kerl's Metallurgy — Copper and Iron 8vo. 

* " '• Steel. Fuel, etc Svo, 

Kunbardt's Ore Dressing in Europe Svo, 

Metcalf's Sieel — A Manual for Steel Users 12mo, 

O'Driscoll's Treatment of Gold Ores 8vo, 

Thurston's Iron and Steel 8vo, 

" Alloys 8vo, 

Wilson's Cyanide Processes 12mo, 

MINERALOGY AND MINING. 

Mine Accidents — Ventilation — Ore Dressing, Etc. 

Barriuger's Minerals of Commercial Value oblong morocco, 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Boyd's Resources of South Western Virginia Svo, 3 00 

" Map of South Western Virginia Pocket-book form, 2 00 

Brush and Penfield's Determinative Mineralogy Svo, 3 50 

Chester's Catalogue of Minerals 8vo, 1 25 

" " paper, 50 

•* Dictionary of tbe Names of Minerals 8vo, 3 00 

Dana's American Localities of Minerals 8vo, 1 00 

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Dana's Descriptive Mineralogy (E. S.) • • • -8vo, half morocco, $12 50 

Mineralogy and Petrography (J.D.) 12mo, 2 00 

" Minerals and How to Study Them. (E. S.) 12mo, 1 50 

" Text-book of Mineralogy. (E. S.) 8vo, 3 50 

^Drinker's Tunnelling, Explosives, Compounds, and Rock Drills. 

4to, half morocco, 25 00 

Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50 

Eissler's Explosives — Nitroglycerine and Dynamite 8vo, 4 00 

Goodyear's Coal Mines of the Western Coast 12mo, 2 50 

Hussak's Rock forming Minerals (Smith.) 8vo, 2 00 

Ihlseng's Manual of Mining . . . . 8vo, 4 00 

Kunhardt's Ore Dressing in Europe «■ 8vo. 1 50 

O'Driscoll's Treatment of Gold Ores . . 8vo, 2 00 

Rosenbusch's Microscopical Physiography of Minerals and 

Rocks (Iddings ) . ., 8vo. 5 00 

Sawyer's Accidents in Mines '. 8vo v 7 00 

Stockbridge's Rocks and Soils. . — .....,.«.. ,. , . .8vo, 2 50 

Walke's Lectures on Explosives 8vo, 4 00 

Williams's Lithology 8vo, 3 00 

Wilson's Mine Ventilation l6mo s 125 

" Hydraulic and Placer Mining 12mo. 

STEAM AND ELECTRICAL ENGINES, BOILERS, Etc. 

Stationaky— Marine— Locomotive — Gas Engines, Etc. 

(See also Engineering, p. 6.) 

Baldwin's Steam Heating for Buildings 12mo, 

Clerk's Gas Engine <. 12mo : 

Ford's Boiler Making for Boiler Makers 18mo, 

Hemen way's Indicator Practice 12mo, 

Hoadley's Warm-blast Furnace 8vo, 

Kneass's Practice and Theory of the Injector 8vo, 

MacCord's Slide Valve 8vo, 

* Maw's Marine Engines Folio, half morocco, 

Meyer's Modern Locomotive Construction 4to, 

Peabody and Miller's Steam Boilers 8vo, 

Peabody's Tables of Saturated Steam 8vo, 

" Thermodynamics of the Steam Engine 8vo, 

" Valve Gears for the Steam Engine 8vo, 

Pray's Twenty Years with the Indicator Royal 8vo, 

Pupin and Osterberg's Thermodynamics 12mo, 

Reagan's Steam and Electrical Locomotives 12mo, 

Rontgen's Thermodynamics. (Du Bois.) 8vo, 

Sinclair's Locomotive Running 12mo, 

Thurston's Boiler Explosion 12mo, 

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Thurston's Engine and Boiler Trials 8vo, $5 00 

" Manual of the Steam Engine. Part I., Structure 

and Theory 8vo, 7 50 

Manual of the Steam Engine. Part II., Design, 

Construction, and Operation 8vo, 7 50 

2 parts, 12 00 

" Philosophy of the Steam Engine 12mo, 75 

" Reflection on the Motive Power of Heat. (Carnot.) 

12mo, 1 50 

Stationary Steam Engines 12mo, 1 50 

" Steam-boiler Construction and Operation 8vo, 5 00 

Spangler's Valve Gears 8vo, 2 50 

Trowbridge's Stationary Steam Engines 4to, boards, 2 50 

Weisbach's Steam Engine. (Du Bois.) 8vo, 5 00 

Whitham's Constructive Steam Engineering 8vo, 10 00 

Steam-engine Design 8vo, 5 00 

Wilson's Steam Boilers. (Flather.) 12mo, 2 50 

Wood's Thermodynamics, Heat Motors, etc 8vo, 4 00 

TABLES, WEIGHTS, AND MEASURES. 

For Actuaries, Chemists, Engineers, Mechanics— Metric 
Tables, Etc. 

Adrian ce's Laboratory Calculations 12mo, 1 25 

Allen's Tables for Iron Analysis 8vo, 3 00 

Bixby's Graphical Computing Tables Sheet, 25 

Compton's Logarithms 12mo, 1 50 

Crandall's Railway and Earthwork Tables 8vo, 1 50 

Egleston's Weights and Measures 18mo, 75 

Fisher's Table of Cubic Yards Cardboard, 25 

Hudson's Excavation Tables. Vol. II 8vo, 1 00 

Johnson's Stadia and Earthwork Tables 8vo, 1 25 

Ludlow's Logarithmic and Other Tables. (Bass.) 12mo, 2 00 

Thurston's Conversion Tables 8vo, 1 00 

Totten's Metrology 8vo, 2 50 

VENTILATION. 

Steam Heating — House Inspection — Mine Ventilation. 

Baldwin's Steam Heating 12mo. 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Carpenter's Heating and Ventilating of Buildings 8vo ; 3 00 

Gerhard's Sanitary House Inspection Square 16mo, 1 00 

Mott's The Air We Breathe, and Ventilation 16mo, 1 00 

Reid's Ventilation of American Dwellings 12mo, 1 50 

Wilson's Mine Ventilation 16mo, 1 25 

15 



MISCELLANEOUS PUBLICATIONS. 

Alcott's Gems, Sentiment, Language Gilt edges, $5 00 

Bailey's The New Tale of a Tub , 8vo, 75 

Ballard's Solution of the Pyramid Problem 8vo, 1 50 

Barnard's The Metrological System of the Great Pyramid. ,8vo. 1 50 

Davis's Elements of Law 8vo, 2 00 

Emmon's Geological Guide-book of the Rocky Mountains. .8vo, 1 50 

Ferrel's Treatise on the Winds 8vo, 4 00 

Haines's Addresses Delivered before'the Am. Ry. Assn. ..12mo. 2 50 

Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 1 00 

Perkins's Cornell University Oblong 4to, 1 50 

Ricketts's History of Rensselaer Polytechnic Institute. . . . 8vo, 3 00 
Rctherham's The New Testament Critically Emphasized. 

12mo, 1 50 
" The Emphasized New Test. A new translation. 

Large 8vo, 2 00 

Totteu's An Important Question in Metrology 8vo, 2 50 

Whitehouse's Lake Mceris „ Paper, 25 

* Wiley's Yosemite, Alaska, and Yellowstone 4to, 3 00 

HEBREW AND CHALDEE TEXT=BOOKS. 

For Schools and Theological Seminaries. 

Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 

(Tregelles.) Small 4to, half morocco, 5 00 

Green's Elementary Hebrew Grammar 12mo, 1 25 

Grammar of the Hebrew Language (New Edition ).8vo, 3 00 

" Hebrew Chrestomathy 8vo, 2 00 

Letteris's Hebrew Bible (Massoretic Notes in English). 

8vo ; arabesque, 2 25 
Luzzato's Grammar of the Biblical Chaldaic Language and the 

Talmud Babli Idioms 12mo, 1 50 

MEDICAL. 

Bull's Maternal Management in Health and Disease 12mo, 1 00 

Hammarsten's Physiological Chemistry. (Maudel.) 8vo, 4 00 

Mott's Composition, Digestibility, and Nutritive Value of Food. 

Large mounted chart. 1 25 

Ruddiman's Incompatibilities in Prescriptions. 8vo, 2 00 

Steel's Treatise on the Diseases of the Ox 8vo, 6 00 

Treatise on the Diseases of the Dog 8vo, 3 50 

Woodhull's Military Hygiene 12mo, 1 50 

Worcester's Small Hospitals — Establishment and Maintenance, 
including Atkinson's Suggestions for Hospital Archi- 
tecture 12mo, 1 25 

16 



